忘记了密码?据/a>新用户?据一种href="//www.parkandroid.com/account/signup/?signup=true&next=/wiki/dedekind-cuts/" id="problem-signup-link-alternative" class="btn-link ax-click" data-ax-id="clicked_signup_from_generic_modal" data-ax-type="button" data-next="/wiki/dedekind-cuts/">注册据/a>
现有的用户?据一种href="//www.parkandroid.com/account/login/?next=/wiki/dedekind-cuts/" id="problem-login-link-alternative" class="btn-link ax-click" data-ax-id="clicked_login_from_generic_modal" data-ax-type="button" data-is_modal="true" data-next="/wiki/dedekind-cuts/">登录据/a>
已经有账户了?据一种href="//www.parkandroid.com/account/login/?next=/wiki/dedekind-cuts/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">这里登录。据/a>
如我们所见,布景据S.pan class="katex"> 问:据/mi> \ mathbb {Q}据/annotation> 问:据/span>的据一种href="//www.parkandroid.com/wiki/rational-numbers/" class="wiki_link" title="有理数" target="_blank">有理数据/a>包含数字的空白如据S.pan class="katex"> 2据/mn> \ sqrt {2}据/annotation> 2据/span> 和据S.pan class="katex"> π据/mi> .据/mi> \π。据/annotation> π据/span>.据/span>我们之前已经获得了一些初步洞察力据一种href="//www.parkandroid.com/wiki/real-numbers/" class="wiki_link" title="实数" target="_blank">实数据/a>这些数字如何允许我们填补空白。在本节中,我们给出了实数的严格数学建设。据/p>
一种据S.trong>有理数分割据/strong> X据/mi> =据/mo> (据/mo> L.据/mi> 那据/mo> 你据/mi> )据/mo> x =(l,u)据/annotation> X据/span>=据/span>(据/span>L.据/span>那据/span>你据/span>)据/span>在据S.pan class="katex"> 问:据/mi> \ mathbb {Q}据/annotation> 问:据/span>是一对子集据S.pan class="katex"> L.据/mi> 那据/mo> 你据/mi> 鲁据/annotation> L.据/span>那据/span>你据/span>的据S.pan class="katex"> 问:据/mi> \ mathbb {Q}据/annotation> 问:据/span>满足以下内容:据/p> L.据/mi> ∪据/mo> 你据/mi> =据/mo> 问:据/mi> 那据/mo> L.据/mi> ∩据/mo> 你据/mi> =据/mo> ∅据/mi> 那据/mo> L.据/mi> ≠据/mo> ∅据/mi> 那据/mo> 你据/mi> ≠据/mo> ∅据/mi> .据/mi> l \ cup u = \ mathbb {q},l \ cap u = \ imptyset,l \ not = \ imptyset,u \ not = \ amptyset。据/annotation> L.据/span>∪据/span>你据/span>=据/span>问:据/span>那据/span>L.据/span>∩据/span>你据/span>=据/span>∅据/span>那据/span>L.据/span>据/span>=据/span>∅据/span>那据/span>你据/span>据/span>=据/span>∅据/span>.据/span> 如果据S.pan class="katex"> L.据/mi> ∈据/mo> L.据/mi> 在l l \据/annotation> L.据/span>∈据/span>L.据/span>和据S.pan class="katex"> 你据/mi> ∈据/mo> 你据/mi> 你在你据/annotation> 你据/span>∈据/span>你据/span>, 然后据S.pan class="katex"> L.据/mi> 据据/mo> 你据/mi> .据/mi> l < u。据/annotation> L.据/span>据据/span>你据/span>.据/span> L.据/mi> L.据/annotation> L.据/span>不包含最大的元素。据/li> 一种据S.trong>实数据/strong>是一种削减的dedekind据S.pan class="katex"> 问:据/mi> \ mathbb {Q}据/annotation> 问:据/span>表示实数的集合据S.pan class="katex"> R.据/mi> \ mathbb {r}据/annotation> R.据/span>.据/p>
一种据S.trong>实数据/strong>是一种削减的dedekind据S.pan class="katex"> 问:据/mi> \ mathbb {Q}据/annotation> 问:据/span>表示实数的集合据S.pan class="katex"> R.据/mi> \ mathbb {r}据/annotation> R.据/span>.据/p>
注意这个切口是据em>订购据/em>和元素的据S.pan class="katex"> L.据/mi> L.据/annotation> L.据/span>(较低)均小于元素据S.pan class="katex"> 你据/mi> 你据/annotation> 你据/span>(如上所述)。在上述定义中,切割据S.pan class="katex"> X据/mi> =据/mo> (据/mo> L.据/mi> 那据/mo> 你据/mi> )据/mo> 那据/mo> x = (L, U),据/annotation> X据/span>=据/span>(据/span>L.据/span>那据/span>你据/span>)据/span>那据/span>我们有据S.pan class="katex"> L.据/mi> =据/mo> 问:据/mi> \据/mi> 你据/mi> l = \ mathbb {q} \ backslash u据/annotation> L.据/span>=据/span>问:据/span>\据/span>你据/span>.给定任意有理数据S.pan class="katex"> R.据/mi> 那据/mo> r,据/annotation> R.据/span>那据/span>切割的一个例子就是其中一种形式据/p> [据/mo> R.据/mi> ]据/mo> =据/mo> (据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> 据据/mo> R.据/mi> }据/mo> 那据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≥据/mo> R.据/mi> }据/mo> )据/mo> 那据/mo> [r] =大(\ \{问\ \ mathbb {q}: q < r \} \{问\ \ mathbb {q}:问\ r组\}\大),据/annotation> [据/span>R.据/span>]据/span>=据/span>(据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>据据/span>R.据/span>}据/span>那据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≥据/span>R.据/span>}据/span>)据/span>那据/span> 它被称为据S.trong>理性的切割据/strong>.这给出了理性数字作为削减的解释,因此每个理性数字也是一个实数。实数为零被定义为理性切割据/p> [据/mo> 0.据/mn> ]据/mo> =据/mo> (据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> 据据/mo> 0.据/mn> }据/mo> 那据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≥据/mo> 0.据/mn> }据/mo> )据/mo> .据/mi> [0] = \ big(\ {q \ in \ mathbb {q}:q <0 \},\ {q \ in \ mathbb {q}:q \ geq 0 \} \ big)。据/annotation> [据/span>0.据/span>]据/span>=据/span>(据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>据据/span>0.据/span>}据/span>那据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≥据/span>0.据/span>}据/span>)据/span>.据/span> 注意在理性的裁剪中据S.pan class="katex"> [据/mo> R.据/mi> ]据/mo> [r]据/annotation> [据/span>R.据/span>]据/span>,一组据S.pan class="katex"> 你据/mi> 你据/annotation> 你据/span>包含一个最小的元素,即据S.pan class="katex"> R.据/mi> R.据/annotation> R.据/span>.实数是据S.trong>非理性的据/strong>如果是集据S.pan class="katex"> 你据/mi> 你据/annotation> 你据/span>不包含最小的元素。定义非理性数的剪切示例是据/p> (据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> 2据/mn> 据据/mo> 2据/mn> }据/mo> 那据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 问:据/mi> 2据/mn> >据/mo> 2据/mn> }据/mo> )据/mo> .据/mi> \ big(\ {q \ in \ mathbb {q}:q \ leq 0 \ text {或} q ^ 2 <2 \},\ {q \ in \ mathbb {q}:q> 0 \ text {and}q ^ 2> 2 \} \ big)。据/annotation> (据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>2据/span>据据/span>2据/span>}据/span>那据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>2据/span>>据/span>2据/span>}据/span>)据/span>.据/span> 我们现在使用削减来定义算术运算和订单关系。据/p>
[据/mo> R.据/mi> ]据/mo> =据/mo> (据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> 据据/mo> R.据/mi> }据/mo> 那据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≥据/mo> R.据/mi> }据/mo> )据/mo> 那据/mo> [r] =大(\ \{问\ \ mathbb {q}: q < r \} \{问\ \ mathbb {q}:问\ r组\}\大),据/annotation> [据/span>R.据/span>]据/span>=据/span>(据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>据据/span>R.据/span>}据/span>那据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≥据/span>R.据/span>}据/span>)据/span>那据/span>
它被称为据S.trong>理性的切割据/strong>.这给出了理性数字作为削减的解释,因此每个理性数字也是一个实数。实数为零被定义为理性切割据/p>
[据/mo> 0.据/mn> ]据/mo> =据/mo> (据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> 据据/mo> 0.据/mn> }据/mo> 那据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≥据/mo> 0.据/mn> }据/mo> )据/mo> .据/mi> [0] = \ big(\ {q \ in \ mathbb {q}:q <0 \},\ {q \ in \ mathbb {q}:q \ geq 0 \} \ big)。据/annotation> [据/span>0.据/span>]据/span>=据/span>(据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>据据/span>0.据/span>}据/span>那据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≥据/span>0.据/span>}据/span>)据/span>.据/span>
注意在理性的裁剪中据S.pan class="katex"> [据/mo> R.据/mi> ]据/mo> [r]据/annotation> [据/span>R.据/span>]据/span>,一组据S.pan class="katex"> 你据/mi> 你据/annotation> 你据/span>包含一个最小的元素,即据S.pan class="katex"> R.据/mi> R.据/annotation> R.据/span>.实数是据S.trong>非理性的据/strong>如果是集据S.pan class="katex"> 你据/mi> 你据/annotation> 你据/span>不包含最小的元素。定义非理性数的剪切示例是据/p> (据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> 2据/mn> 据据/mo> 2据/mn> }据/mo> 那据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 问:据/mi> 2据/mn> >据/mo> 2据/mn> }据/mo> )据/mo> .据/mi> \ big(\ {q \ in \ mathbb {q}:q \ leq 0 \ text {或} q ^ 2 <2 \},\ {q \ in \ mathbb {q}:q> 0 \ text {and}q ^ 2> 2 \} \ big)。据/annotation> (据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>2据/span>据据/span>2据/span>}据/span>那据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>2据/span>>据/span>2据/span>}据/span>)据/span>.据/span> 我们现在使用削减来定义算术运算和订单关系。据/p>
(据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> 2据/mn> 据据/mo> 2据/mn> }据/mo> 那据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 问:据/mi> 2据/mn> >据/mo> 2据/mn> }据/mo> )据/mo> .据/mi> \ big(\ {q \ in \ mathbb {q}:q \ leq 0 \ text {或} q ^ 2 <2 \},\ {q \ in \ mathbb {q}:q> 0 \ text {and}q ^ 2> 2 \} \ big)。据/annotation> (据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>2据/span>据据/span>2据/span>}据/span>那据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>2据/span>>据/span>2据/span>}据/span>)据/span>.据/span>
我们现在使用削减来定义算术运算和订单关系。据/p>
我们首先使用切来定义实数熟悉的算术运算,包括加、减、乘和除。削减据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> B.据/mi> )据/mo> x = (A, B)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>B.据/span>)据/span>和据S.pan class="katex"> y据/mi> =据/mo> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> y = (C, D)据/annotation> y据/span>=据/span>(据/span>C据/span>那据/span>D.据/span>)据/span>,定义添加如下:据/p> X据/mi> +据/mo> y据/mi> =据/mo> (据/mo> E.据/mi> 那据/mo> 问:据/mi> \据/mi> E.据/mi> )据/mo> 那据/mo> 在哪里据/mtext> E.据/mi> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> =据/mo> 一种据/mi> +据/mo> C据/mi> 对于一些据/mtext> 一种据/mi> ∈据/mo> 一种据/mi> 那据/mo> C据/mi> ∈据/mo> C据/mi> }据/mo> .据/mi> x + y =(e,\ mathbb {q} \ backslash e),\ text {where} e = \ {q \ in \ mathbb {q}:q = a + c \ text {for some} a \,c \在c \}中。据/annotation> X据/span>+据/span>y据/span>=据/span>(据/span>E.据/span>那据/span>问:据/span>\据/span>E.据/span>)据/span>那据/span>在哪里据/span>E.据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>=据/span>一种据/span>+据/span>C据/span>对于一些据/span>一种据/span>∈据/span>一种据/span>那据/span>C据/span>∈据/span>C据/span>}据/span>.据/span> 观察到削减添加是明确的,满足据S.pan class="katex"> X据/mi> +据/mo> [据/mo> 0.据/mn> ]据/mo> =据/mo> X据/mi> 那据/mo> X据/mi> +据/mo> y据/mi> =据/mo> y据/mi> +据/mo> X据/mi> x + [0] = x,x + y = y + x据/annotation> X据/span>+据/span>[据/span>0.据/span>]据/span>=据/span>X据/span>那据/span>X据/span>+据/span>y据/span>=据/span>y据/span>+据/span>X据/span>(换向),和据S.pan class="katex"> (据/mo> X据/mi> +据/mo> y据/mi> )据/mo> +据/mo> Z.据/mi> =据/mo> X据/mi> +据/mo> (据/mo> y据/mi> +据/mo> Z.据/mi> )据/mo> (x + y)+ z = x +(y + z)据/annotation> (据/span>X据/span>+据/span>y据/span>)据/span>+据/span>Z.据/span>=据/span>X据/span>+据/span>(据/span>y据/span>+据/span>Z.据/span>)据/span>(结合性)。的加性逆据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> B.据/mi> )据/mo> x = (A, B)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>B.据/span>)据/span>是据S.pan class="katex"> -据/mo> X据/mi> =据/mo> (据/mo> C据/mi> 那据/mo> 问:据/mi> \据/mi> C据/mi> )据/mo> -x =(c,\ mathbb {q} \ backslash c)据/annotation> -据/span>X据/span>=据/span>(据/span>C据/span>那据/span>问:据/span>\据/span>C据/span>)据/span>和据/p> C据/mi> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> =据/mo> -据/mo> B.据/mi> 为了据/mtext> B.据/mi> ∈据/mo> B.据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> c = \ {q \ in \ mathbb {q}:q = -b \ text {b \ text {not最小元素} \}。据/annotation> C据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>=据/span>-据/span>B.据/span>为了据/span>B.据/span>∈据/span>B.据/span>不是最小的元素据/span>}据/span>.据/span> 然后减法由据S.pan class="katex"> X据/mi> -据/mo> y据/mi> =据/mo> X据/mi> +据/mo> (据/mo> -据/mo> y据/mi> )据/mo> 那据/mo> X -y = X + (-y)据/annotation> X据/span>-据/span>y据/span>=据/span>X据/span>+据/span>(据/span>-据/span>y据/span>)据/span>那据/span>和任何削减据S.pan class="katex"> X据/mi> X据/annotation> X据/span>, 我们有据S.pan class="katex"> X据/mi> +据/mo> (据/mo> -据/mo> X据/mi> )据/mo> =据/mo> (据/mo> -据/mo> X据/mi> )据/mo> +据/mo> X据/mi> =据/mo> [据/mo> 0.据/mn> ]据/mo> .据/mi> X + (-x) = (-x) + X =[0]。据/annotation> X据/span>+据/span>(据/span>-据/span>X据/span>)据/span>=据/span>(据/span>-据/span>X据/span>)据/span>+据/span>X据/span>=据/span>[据/span>0.据/span>]据/span>.据/span>绝对值据S.pan class="katex"> |据/mi> X据/mi> |据/mi> \ vert x \ vert据/annotation> |据/span>X据/span>|据/span>是由的据/p> |据/mo> X据/mi> |据/mo> =据/mo> {据/mo> X据/mi> 如果据/mtext> X据/mi> ≥据/mo> 0.据/mn> -据/mo> X据/mi> 否则据/mtext> 那据/mo> \ Levert X \ Rvert = \ Begin {案例} x&\ mbox {if} x \ geq 0 \\ -x&\ mbox {否则},\结束{案例}据/annotation> |据/span>X据/span>|据/span>=据/span>{据/span>X据/span>-据/span>X据/span>如果据/span>X据/span>≥据/span>0.据/span>否则据/span>那据/span> 并乘法定义据/p> X据/mi> ×据/mo> y据/mi> =据/mo> {据/mo> 0.据/mn> 如果据/mtext> X据/mi> =据/mo> 0.据/mn> 或据/mtext> y据/mi> =据/mo> 0.据/mn> |据/mo> X据/mi> |据/mo> ×据/mo> |据/mo> y据/mi> |据/mo> 如果据/mtext> X据/mi> >据/mo> 0.据/mn> 那据/mo> y据/mi> >据/mo> 0.据/mn> 或据/mtext> X据/mi> 据据/mo> 0.据/mn> 那据/mo> y据/mi> 据据/mo> 0.据/mn> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> ×据/mo> |据/mo> y据/mi> |据/mo> )据/mo> 如果据/mtext> X据/mi> >据/mo> 0.据/mn> 那据/mo> y据/mi> 据据/mo> 0.据/mn> 或据/mtext> X据/mi> 据据/mo> 0.据/mn> 那据/mo> y据/mi> >据/mo> 0.据/mn> x \ times y = \ begin {uis} 0&\ mbox {if} x = 0 \ mbox {或} y = 0 \\ \ ltvert x \ rfter \ times \ ltvert y \ rfter&\ mbox {if} x>0,Y> 0 \ \ mbox {或} \ x <0,y <0 \\ - \ big(\ ltvert x \ rfter \ times \ ltvert y \ rvert \ big)&\ mbox {if} x> 0,y <0 \ \ mbox {或} \ x <0,Y> 0. \ END {案例}据/annotation> X据/span>×据/span>y据/span>=据/span>⎩据/span>⎪据/span>⎨据/span>⎪据/span>⎧据/span>0.据/span>|据/span>X据/span>|据/span>×据/span>|据/span>y据/span>|据/span>-据/span>(据/span>|据/span>X据/span>|据/span>×据/span>|据/span>y据/span>|据/span>)据/span>如果据/span>X据/span>=据/span>0.据/span>或据/span>y据/span>=据/span>0.据/span>如果据/span>X据/span>>据/span>0.据/span>那据/span>y据/span>>据/span>0.据/span>或据/span>X据/span>据据/span>0.据/span>那据/span>y据/span>据据/span>0.据/span>如果据/span>X据/span>>据/span>0.据/span>那据/span>y据/span>据据/span>0.据/span>或据/span>X据/span>据据/span>0.据/span>那据/span>y据/span>>据/span>0.据/span>.据/span> 然后加法和乘法满足属性据S.pan class="katex"> X据/mi> ×据/mo> [据/mo> 1据/mn> ]据/mo> =据/mo> X据/mi> 那据/mo> X据/mi> ×据/mo> y据/mi> =据/mo> y据/mi> ×据/mo> X据/mi> x \次[1] = x,x \ times y = y \ times x据/annotation> X据/span>×据/span>[据/span>1据/span>]据/span>=据/span>X据/span>那据/span>X据/span>×据/span>y据/span>=据/span>y据/span>×据/span>X据/span>(交换性),据S.pan class="katex"> (据/mo> X据/mi> ×据/mo> y据/mi> )据/mo> ×据/mo> Z.据/mi> =据/mo> X据/mi> ×据/mo> (据/mo> y据/mi> ×据/mo> Z.据/mi> )据/mo> (x \乘以y) \乘以z = x \乘以(y \乘以z)据/annotation> (据/span>X据/span>×据/span>y据/span>)据/span>×据/span>Z.据/span>=据/span>X据/span>×据/span>(据/span>y据/span>×据/span>Z.据/span>)据/span>(关联),和据S.pan class="katex"> (据/mo> X据/mi> +据/mo> y据/mi> )据/mo> ×据/mo> Z.据/mi> =据/mo> (据/mo> X据/mi> ×据/mo> Z.据/mi> )据/mo> +据/mo> (据/mo> y据/mi> ×据/mo> Z.据/mi> )据/mo> (x + y) \ * z = (x \ * z) + (y \ * z)据/annotation> (据/span>X据/span>+据/span>y据/span>)据/span>×据/span>Z.据/span>=据/span>(据/span>X据/span>×据/span>Z.据/span>)据/span>+据/span>(据/span>y据/span>×据/span>Z.据/span>)据/span>(分配性)。据/p> 对于实数据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> )据/mo> x = (A, \mathbb{Q} \反斜杠A)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>问:据/span>\据/span>一种据/span>)据/span>和据S.pan class="katex"> X据/mi> >据/mo> 0.据/mn> x > 0据/annotation> X据/span>>据/span>0.据/span>,乘法逆定义据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> (据/mo> 一种据/mi> '据/mo> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> '据/mo> )据/mo> x^{-1} = (A', \mathbb{Q} \反斜杠A')据/annotation> X据/span>-据/span>1据/span>=据/span>(据/span>一种据/span>'据/span>那据/span>问:据/span>\据/span>一种据/span>'据/span>)据/span>, 在哪里据/p> 一种据/mi> '据/mo> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 1据/mn> 问:据/mi> ∈据/mo> 问:据/mi> \据/mi> 一种据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> a'= \ left \ {q \ in \ mathbb {q}:q \ leq 0 \ mbox {或} q> 0 \\ \ mbox {and} \ frac {1} {q} \ in \ mathbb {q}\ backslash a \\ \ text {不是最小的元素} \ \ \}。据/annotation> 一种据/span>'据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>1据/span>∈据/span>问:据/span>\据/span>一种据/span>不是最小的元素据/span>}据/span>.据/span> 为了据S.pan class="katex"> X据/mi> 据据/mo> 0.据/mn> x < 0据/annotation> X据/span>据据/span>0.据/span>那据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> -据/mo> 1据/mn> )据/mo> x ^ { - 1} = - \左(\ ltvert x \ rvert ^ { - 1}右)据/annotation> X据/span>-据/span>1据/span>=据/span>-据/span>(据/span>|据/span>X据/span>|据/span>-据/span>1据/span>)据/span>.据/p> 对于Rational Cuts,所有上述算术运算都与Rational的算术运算一致。据/p>
X据/mi> +据/mo> y据/mi> =据/mo> (据/mo> E.据/mi> 那据/mo> 问:据/mi> \据/mi> E.据/mi> )据/mo> 那据/mo> 在哪里据/mtext> E.据/mi> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> =据/mo> 一种据/mi> +据/mo> C据/mi> 对于一些据/mtext> 一种据/mi> ∈据/mo> 一种据/mi> 那据/mo> C据/mi> ∈据/mo> C据/mi> }据/mo> .据/mi> x + y =(e,\ mathbb {q} \ backslash e),\ text {where} e = \ {q \ in \ mathbb {q}:q = a + c \ text {for some} a \,c \在c \}中。据/annotation> X据/span>+据/span>y据/span>=据/span>(据/span>E.据/span>那据/span>问:据/span>\据/span>E.据/span>)据/span>那据/span>在哪里据/span>E.据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>=据/span>一种据/span>+据/span>C据/span>对于一些据/span>一种据/span>∈据/span>一种据/span>那据/span>C据/span>∈据/span>C据/span>}据/span>.据/span>
观察到削减添加是明确的,满足据S.pan class="katex"> X据/mi> +据/mo> [据/mo> 0.据/mn> ]据/mo> =据/mo> X据/mi> 那据/mo> X据/mi> +据/mo> y据/mi> =据/mo> y据/mi> +据/mo> X据/mi> x + [0] = x,x + y = y + x据/annotation> X据/span>+据/span>[据/span>0.据/span>]据/span>=据/span>X据/span>那据/span>X据/span>+据/span>y据/span>=据/span>y据/span>+据/span>X据/span>(换向),和据S.pan class="katex"> (据/mo> X据/mi> +据/mo> y据/mi> )据/mo> +据/mo> Z.据/mi> =据/mo> X据/mi> +据/mo> (据/mo> y据/mi> +据/mo> Z.据/mi> )据/mo> (x + y)+ z = x +(y + z)据/annotation> (据/span>X据/span>+据/span>y据/span>)据/span>+据/span>Z.据/span>=据/span>X据/span>+据/span>(据/span>y据/span>+据/span>Z.据/span>)据/span>(结合性)。的加性逆据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> B.据/mi> )据/mo> x = (A, B)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>B.据/span>)据/span>是据S.pan class="katex"> -据/mo> X据/mi> =据/mo> (据/mo> C据/mi> 那据/mo> 问:据/mi> \据/mi> C据/mi> )据/mo> -x =(c,\ mathbb {q} \ backslash c)据/annotation> -据/span>X据/span>=据/span>(据/span>C据/span>那据/span>问:据/span>\据/span>C据/span>)据/span>和据/p> C据/mi> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> =据/mo> -据/mo> B.据/mi> 为了据/mtext> B.据/mi> ∈据/mo> B.据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> c = \ {q \ in \ mathbb {q}:q = -b \ text {b \ text {not最小元素} \}。据/annotation> C据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>=据/span>-据/span>B.据/span>为了据/span>B.据/span>∈据/span>B.据/span>不是最小的元素据/span>}据/span>.据/span> 然后减法由据S.pan class="katex"> X据/mi> -据/mo> y据/mi> =据/mo> X据/mi> +据/mo> (据/mo> -据/mo> y据/mi> )据/mo> 那据/mo> X -y = X + (-y)据/annotation> X据/span>-据/span>y据/span>=据/span>X据/span>+据/span>(据/span>-据/span>y据/span>)据/span>那据/span>和任何削减据S.pan class="katex"> X据/mi> X据/annotation> X据/span>, 我们有据S.pan class="katex"> X据/mi> +据/mo> (据/mo> -据/mo> X据/mi> )据/mo> =据/mo> (据/mo> -据/mo> X据/mi> )据/mo> +据/mo> X据/mi> =据/mo> [据/mo> 0.据/mn> ]据/mo> .据/mi> X + (-x) = (-x) + X =[0]。据/annotation> X据/span>+据/span>(据/span>-据/span>X据/span>)据/span>=据/span>(据/span>-据/span>X据/span>)据/span>+据/span>X据/span>=据/span>[据/span>0.据/span>]据/span>.据/span>绝对值据S.pan class="katex"> |据/mi> X据/mi> |据/mi> \ vert x \ vert据/annotation> |据/span>X据/span>|据/span>是由的据/p> |据/mo> X据/mi> |据/mo> =据/mo> {据/mo> X据/mi> 如果据/mtext> X据/mi> ≥据/mo> 0.据/mn> -据/mo> X据/mi> 否则据/mtext> 那据/mo> \ Levert X \ Rvert = \ Begin {案例} x&\ mbox {if} x \ geq 0 \\ -x&\ mbox {否则},\结束{案例}据/annotation> |据/span>X据/span>|据/span>=据/span>{据/span>X据/span>-据/span>X据/span>如果据/span>X据/span>≥据/span>0.据/span>否则据/span>那据/span> 并乘法定义据/p> X据/mi> ×据/mo> y据/mi> =据/mo> {据/mo> 0.据/mn> 如果据/mtext> X据/mi> =据/mo> 0.据/mn> 或据/mtext> y据/mi> =据/mo> 0.据/mn> |据/mo> X据/mi> |据/mo> ×据/mo> |据/mo> y据/mi> |据/mo> 如果据/mtext> X据/mi> >据/mo> 0.据/mn> 那据/mo> y据/mi> >据/mo> 0.据/mn> 或据/mtext> X据/mi> 据据/mo> 0.据/mn> 那据/mo> y据/mi> 据据/mo> 0.据/mn> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> ×据/mo> |据/mo> y据/mi> |据/mo> )据/mo> 如果据/mtext> X据/mi> >据/mo> 0.据/mn> 那据/mo> y据/mi> 据据/mo> 0.据/mn> 或据/mtext> X据/mi> 据据/mo> 0.据/mn> 那据/mo> y据/mi> >据/mo> 0.据/mn> x \ times y = \ begin {uis} 0&\ mbox {if} x = 0 \ mbox {或} y = 0 \\ \ ltvert x \ rfter \ times \ ltvert y \ rfter&\ mbox {if} x>0,Y> 0 \ \ mbox {或} \ x <0,y <0 \\ - \ big(\ ltvert x \ rfter \ times \ ltvert y \ rvert \ big)&\ mbox {if} x> 0,y <0 \ \ mbox {或} \ x <0,Y> 0. \ END {案例}据/annotation> X据/span>×据/span>y据/span>=据/span>⎩据/span>⎪据/span>⎨据/span>⎪据/span>⎧据/span>0.据/span>|据/span>X据/span>|据/span>×据/span>|据/span>y据/span>|据/span>-据/span>(据/span>|据/span>X据/span>|据/span>×据/span>|据/span>y据/span>|据/span>)据/span>如果据/span>X据/span>=据/span>0.据/span>或据/span>y据/span>=据/span>0.据/span>如果据/span>X据/span>>据/span>0.据/span>那据/span>y据/span>>据/span>0.据/span>或据/span>X据/span>据据/span>0.据/span>那据/span>y据/span>据据/span>0.据/span>如果据/span>X据/span>>据/span>0.据/span>那据/span>y据/span>据据/span>0.据/span>或据/span>X据/span>据据/span>0.据/span>那据/span>y据/span>>据/span>0.据/span>.据/span> 然后加法和乘法满足属性据S.pan class="katex"> X据/mi> ×据/mo> [据/mo> 1据/mn> ]据/mo> =据/mo> X据/mi> 那据/mo> X据/mi> ×据/mo> y据/mi> =据/mo> y据/mi> ×据/mo> X据/mi> x \次[1] = x,x \ times y = y \ times x据/annotation> X据/span>×据/span>[据/span>1据/span>]据/span>=据/span>X据/span>那据/span>X据/span>×据/span>y据/span>=据/span>y据/span>×据/span>X据/span>(交换性),据S.pan class="katex"> (据/mo> X据/mi> ×据/mo> y据/mi> )据/mo> ×据/mo> Z.据/mi> =据/mo> X据/mi> ×据/mo> (据/mo> y据/mi> ×据/mo> Z.据/mi> )据/mo> (x \乘以y) \乘以z = x \乘以(y \乘以z)据/annotation> (据/span>X据/span>×据/span>y据/span>)据/span>×据/span>Z.据/span>=据/span>X据/span>×据/span>(据/span>y据/span>×据/span>Z.据/span>)据/span>(关联),和据S.pan class="katex"> (据/mo> X据/mi> +据/mo> y据/mi> )据/mo> ×据/mo> Z.据/mi> =据/mo> (据/mo> X据/mi> ×据/mo> Z.据/mi> )据/mo> +据/mo> (据/mo> y据/mi> ×据/mo> Z.据/mi> )据/mo> (x + y) \ * z = (x \ * z) + (y \ * z)据/annotation> (据/span>X据/span>+据/span>y据/span>)据/span>×据/span>Z.据/span>=据/span>(据/span>X据/span>×据/span>Z.据/span>)据/span>+据/span>(据/span>y据/span>×据/span>Z.据/span>)据/span>(分配性)。据/p> 对于实数据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> )据/mo> x = (A, \mathbb{Q} \反斜杠A)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>问:据/span>\据/span>一种据/span>)据/span>和据S.pan class="katex"> X据/mi> >据/mo> 0.据/mn> x > 0据/annotation> X据/span>>据/span>0.据/span>,乘法逆定义据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> (据/mo> 一种据/mi> '据/mo> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> '据/mo> )据/mo> x^{-1} = (A', \mathbb{Q} \反斜杠A')据/annotation> X据/span>-据/span>1据/span>=据/span>(据/span>一种据/span>'据/span>那据/span>问:据/span>\据/span>一种据/span>'据/span>)据/span>, 在哪里据/p> 一种据/mi> '据/mo> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 1据/mn> 问:据/mi> ∈据/mo> 问:据/mi> \据/mi> 一种据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> a'= \ left \ {q \ in \ mathbb {q}:q \ leq 0 \ mbox {或} q> 0 \\ \ mbox {and} \ frac {1} {q} \ in \ mathbb {q}\ backslash a \\ \ text {不是最小的元素} \ \ \}。据/annotation> 一种据/span>'据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>1据/span>∈据/span>问:据/span>\据/span>一种据/span>不是最小的元素据/span>}据/span>.据/span> 为了据S.pan class="katex"> X据/mi> 据据/mo> 0.据/mn> x < 0据/annotation> X据/span>据据/span>0.据/span>那据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> -据/mo> 1据/mn> )据/mo> x ^ { - 1} = - \左(\ ltvert x \ rvert ^ { - 1}右)据/annotation> X据/span>-据/span>1据/span>=据/span>-据/span>(据/span>|据/span>X据/span>|据/span>-据/span>1据/span>)据/span>.据/p> 对于Rational Cuts,所有上述算术运算都与Rational的算术运算一致。据/p>
C据/mi> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> =据/mo> -据/mo> B.据/mi> 为了据/mtext> B.据/mi> ∈据/mo> B.据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> c = \ {q \ in \ mathbb {q}:q = -b \ text {b \ text {not最小元素} \}。据/annotation> C据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>=据/span>-据/span>B.据/span>为了据/span>B.据/span>∈据/span>B.据/span>不是最小的元素据/span>}据/span>.据/span>
然后减法由据S.pan class="katex"> X据/mi> -据/mo> y据/mi> =据/mo> X据/mi> +据/mo> (据/mo> -据/mo> y据/mi> )据/mo> 那据/mo> X -y = X + (-y)据/annotation> X据/span>-据/span>y据/span>=据/span>X据/span>+据/span>(据/span>-据/span>y据/span>)据/span>那据/span>和任何削减据S.pan class="katex"> X据/mi> X据/annotation> X据/span>, 我们有据S.pan class="katex"> X据/mi> +据/mo> (据/mo> -据/mo> X据/mi> )据/mo> =据/mo> (据/mo> -据/mo> X据/mi> )据/mo> +据/mo> X据/mi> =据/mo> [据/mo> 0.据/mn> ]据/mo> .据/mi> X + (-x) = (-x) + X =[0]。据/annotation> X据/span>+据/span>(据/span>-据/span>X据/span>)据/span>=据/span>(据/span>-据/span>X据/span>)据/span>+据/span>X据/span>=据/span>[据/span>0.据/span>]据/span>.据/span>绝对值据S.pan class="katex"> |据/mi> X据/mi> |据/mi> \ vert x \ vert据/annotation> |据/span>X据/span>|据/span>是由的据/p> |据/mo> X据/mi> |据/mo> =据/mo> {据/mo> X据/mi> 如果据/mtext> X据/mi> ≥据/mo> 0.据/mn> -据/mo> X据/mi> 否则据/mtext> 那据/mo> \ Levert X \ Rvert = \ Begin {案例} x&\ mbox {if} x \ geq 0 \\ -x&\ mbox {否则},\结束{案例}据/annotation> |据/span>X据/span>|据/span>=据/span>{据/span>X据/span>-据/span>X据/span>如果据/span>X据/span>≥据/span>0.据/span>否则据/span>那据/span> 并乘法定义据/p> X据/mi> ×据/mo> y据/mi> =据/mo> {据/mo> 0.据/mn> 如果据/mtext> X据/mi> =据/mo> 0.据/mn> 或据/mtext> y据/mi> =据/mo> 0.据/mn> |据/mo> X据/mi> |据/mo> ×据/mo> |据/mo> y据/mi> |据/mo> 如果据/mtext> X据/mi> >据/mo> 0.据/mn> 那据/mo> y据/mi> >据/mo> 0.据/mn> 或据/mtext> X据/mi> 据据/mo> 0.据/mn> 那据/mo> y据/mi> 据据/mo> 0.据/mn> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> ×据/mo> |据/mo> y据/mi> |据/mo> )据/mo> 如果据/mtext> X据/mi> >据/mo> 0.据/mn> 那据/mo> y据/mi> 据据/mo> 0.据/mn> 或据/mtext> X据/mi> 据据/mo> 0.据/mn> 那据/mo> y据/mi> >据/mo> 0.据/mn> x \ times y = \ begin {uis} 0&\ mbox {if} x = 0 \ mbox {或} y = 0 \\ \ ltvert x \ rfter \ times \ ltvert y \ rfter&\ mbox {if} x>0,Y> 0 \ \ mbox {或} \ x <0,y <0 \\ - \ big(\ ltvert x \ rfter \ times \ ltvert y \ rvert \ big)&\ mbox {if} x> 0,y <0 \ \ mbox {或} \ x <0,Y> 0. \ END {案例}据/annotation> X据/span>×据/span>y据/span>=据/span>⎩据/span>⎪据/span>⎨据/span>⎪据/span>⎧据/span>0.据/span>|据/span>X据/span>|据/span>×据/span>|据/span>y据/span>|据/span>-据/span>(据/span>|据/span>X据/span>|据/span>×据/span>|据/span>y据/span>|据/span>)据/span>如果据/span>X据/span>=据/span>0.据/span>或据/span>y据/span>=据/span>0.据/span>如果据/span>X据/span>>据/span>0.据/span>那据/span>y据/span>>据/span>0.据/span>或据/span>X据/span>据据/span>0.据/span>那据/span>y据/span>据据/span>0.据/span>如果据/span>X据/span>>据/span>0.据/span>那据/span>y据/span>据据/span>0.据/span>或据/span>X据/span>据据/span>0.据/span>那据/span>y据/span>>据/span>0.据/span>.据/span> 然后加法和乘法满足属性据S.pan class="katex"> X据/mi> ×据/mo> [据/mo> 1据/mn> ]据/mo> =据/mo> X据/mi> 那据/mo> X据/mi> ×据/mo> y据/mi> =据/mo> y据/mi> ×据/mo> X据/mi> x \次[1] = x,x \ times y = y \ times x据/annotation> X据/span>×据/span>[据/span>1据/span>]据/span>=据/span>X据/span>那据/span>X据/span>×据/span>y据/span>=据/span>y据/span>×据/span>X据/span>(交换性),据S.pan class="katex"> (据/mo> X据/mi> ×据/mo> y据/mi> )据/mo> ×据/mo> Z.据/mi> =据/mo> X据/mi> ×据/mo> (据/mo> y据/mi> ×据/mo> Z.据/mi> )据/mo> (x \乘以y) \乘以z = x \乘以(y \乘以z)据/annotation> (据/span>X据/span>×据/span>y据/span>)据/span>×据/span>Z.据/span>=据/span>X据/span>×据/span>(据/span>y据/span>×据/span>Z.据/span>)据/span>(关联),和据S.pan class="katex"> (据/mo> X据/mi> +据/mo> y据/mi> )据/mo> ×据/mo> Z.据/mi> =据/mo> (据/mo> X据/mi> ×据/mo> Z.据/mi> )据/mo> +据/mo> (据/mo> y据/mi> ×据/mo> Z.据/mi> )据/mo> (x + y) \ * z = (x \ * z) + (y \ * z)据/annotation> (据/span>X据/span>+据/span>y据/span>)据/span>×据/span>Z.据/span>=据/span>(据/span>X据/span>×据/span>Z.据/span>)据/span>+据/span>(据/span>y据/span>×据/span>Z.据/span>)据/span>(分配性)。据/p> 对于实数据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> )据/mo> x = (A, \mathbb{Q} \反斜杠A)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>问:据/span>\据/span>一种据/span>)据/span>和据S.pan class="katex"> X据/mi> >据/mo> 0.据/mn> x > 0据/annotation> X据/span>>据/span>0.据/span>,乘法逆定义据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> (据/mo> 一种据/mi> '据/mo> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> '据/mo> )据/mo> x^{-1} = (A', \mathbb{Q} \反斜杠A')据/annotation> X据/span>-据/span>1据/span>=据/span>(据/span>一种据/span>'据/span>那据/span>问:据/span>\据/span>一种据/span>'据/span>)据/span>, 在哪里据/p> 一种据/mi> '据/mo> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 1据/mn> 问:据/mi> ∈据/mo> 问:据/mi> \据/mi> 一种据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> a'= \ left \ {q \ in \ mathbb {q}:q \ leq 0 \ mbox {或} q> 0 \\ \ mbox {and} \ frac {1} {q} \ in \ mathbb {q}\ backslash a \\ \ text {不是最小的元素} \ \ \}。据/annotation> 一种据/span>'据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>1据/span>∈据/span>问:据/span>\据/span>一种据/span>不是最小的元素据/span>}据/span>.据/span> 为了据S.pan class="katex"> X据/mi> 据据/mo> 0.据/mn> x < 0据/annotation> X据/span>据据/span>0.据/span>那据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> -据/mo> 1据/mn> )据/mo> x ^ { - 1} = - \左(\ ltvert x \ rvert ^ { - 1}右)据/annotation> X据/span>-据/span>1据/span>=据/span>-据/span>(据/span>|据/span>X据/span>|据/span>-据/span>1据/span>)据/span>.据/p> 对于Rational Cuts,所有上述算术运算都与Rational的算术运算一致。据/p>
|据/mo> X据/mi> |据/mo> =据/mo> {据/mo> X据/mi> 如果据/mtext> X据/mi> ≥据/mo> 0.据/mn> -据/mo> X据/mi> 否则据/mtext> 那据/mo> \ Levert X \ Rvert = \ Begin {案例} x&\ mbox {if} x \ geq 0 \\ -x&\ mbox {否则},\结束{案例}据/annotation> |据/span>X据/span>|据/span>=据/span>{据/span>X据/span>-据/span>X据/span>如果据/span>X据/span>≥据/span>0.据/span>否则据/span>那据/span>
并乘法定义据/p>
X据/mi> ×据/mo> y据/mi> =据/mo> {据/mo> 0.据/mn> 如果据/mtext> X据/mi> =据/mo> 0.据/mn> 或据/mtext> y据/mi> =据/mo> 0.据/mn> |据/mo> X据/mi> |据/mo> ×据/mo> |据/mo> y据/mi> |据/mo> 如果据/mtext> X据/mi> >据/mo> 0.据/mn> 那据/mo> y据/mi> >据/mo> 0.据/mn> 或据/mtext> X据/mi> 据据/mo> 0.据/mn> 那据/mo> y据/mi> 据据/mo> 0.据/mn> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> ×据/mo> |据/mo> y据/mi> |据/mo> )据/mo> 如果据/mtext> X据/mi> >据/mo> 0.据/mn> 那据/mo> y据/mi> 据据/mo> 0.据/mn> 或据/mtext> X据/mi> 据据/mo> 0.据/mn> 那据/mo> y据/mi> >据/mo> 0.据/mn> x \ times y = \ begin {uis} 0&\ mbox {if} x = 0 \ mbox {或} y = 0 \\ \ ltvert x \ rfter \ times \ ltvert y \ rfter&\ mbox {if} x>0,Y> 0 \ \ mbox {或} \ x <0,y <0 \\ - \ big(\ ltvert x \ rfter \ times \ ltvert y \ rvert \ big)&\ mbox {if} x> 0,y <0 \ \ mbox {或} \ x <0,Y> 0. \ END {案例}据/annotation> X据/span>×据/span>y据/span>=据/span>⎩据/span>⎪据/span>⎨据/span>⎪据/span>⎧据/span>0.据/span>|据/span>X据/span>|据/span>×据/span>|据/span>y据/span>|据/span>-据/span>(据/span>|据/span>X据/span>|据/span>×据/span>|据/span>y据/span>|据/span>)据/span>如果据/span>X据/span>=据/span>0.据/span>或据/span>y据/span>=据/span>0.据/span>如果据/span>X据/span>>据/span>0.据/span>那据/span>y据/span>>据/span>0.据/span>或据/span>X据/span>据据/span>0.据/span>那据/span>y据/span>据据/span>0.据/span>如果据/span>X据/span>>据/span>0.据/span>那据/span>y据/span>据据/span>0.据/span>或据/span>X据/span>据据/span>0.据/span>那据/span>y据/span>>据/span>0.据/span>.据/span>
然后加法和乘法满足属性据S.pan class="katex"> X据/mi> ×据/mo> [据/mo> 1据/mn> ]据/mo> =据/mo> X据/mi> 那据/mo> X据/mi> ×据/mo> y据/mi> =据/mo> y据/mi> ×据/mo> X据/mi> x \次[1] = x,x \ times y = y \ times x据/annotation> X据/span>×据/span>[据/span>1据/span>]据/span>=据/span>X据/span>那据/span>X据/span>×据/span>y据/span>=据/span>y据/span>×据/span>X据/span>(交换性),据S.pan class="katex"> (据/mo> X据/mi> ×据/mo> y据/mi> )据/mo> ×据/mo> Z.据/mi> =据/mo> X据/mi> ×据/mo> (据/mo> y据/mi> ×据/mo> Z.据/mi> )据/mo> (x \乘以y) \乘以z = x \乘以(y \乘以z)据/annotation> (据/span>X据/span>×据/span>y据/span>)据/span>×据/span>Z.据/span>=据/span>X据/span>×据/span>(据/span>y据/span>×据/span>Z.据/span>)据/span>(关联),和据S.pan class="katex"> (据/mo> X据/mi> +据/mo> y据/mi> )据/mo> ×据/mo> Z.据/mi> =据/mo> (据/mo> X据/mi> ×据/mo> Z.据/mi> )据/mo> +据/mo> (据/mo> y据/mi> ×据/mo> Z.据/mi> )据/mo> (x + y) \ * z = (x \ * z) + (y \ * z)据/annotation> (据/span>X据/span>+据/span>y据/span>)据/span>×据/span>Z.据/span>=据/span>(据/span>X据/span>×据/span>Z.据/span>)据/span>+据/span>(据/span>y据/span>×据/span>Z.据/span>)据/span>(分配性)。据/p> 对于实数据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> )据/mo> x = (A, \mathbb{Q} \反斜杠A)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>问:据/span>\据/span>一种据/span>)据/span>和据S.pan class="katex"> X据/mi> >据/mo> 0.据/mn> x > 0据/annotation> X据/span>>据/span>0.据/span>,乘法逆定义据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> (据/mo> 一种据/mi> '据/mo> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> '据/mo> )据/mo> x^{-1} = (A', \mathbb{Q} \反斜杠A')据/annotation> X据/span>-据/span>1据/span>=据/span>(据/span>一种据/span>'据/span>那据/span>问:据/span>\据/span>一种据/span>'据/span>)据/span>, 在哪里据/p> 一种据/mi> '据/mo> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 1据/mn> 问:据/mi> ∈据/mo> 问:据/mi> \据/mi> 一种据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> a'= \ left \ {q \ in \ mathbb {q}:q \ leq 0 \ mbox {或} q> 0 \\ \ mbox {and} \ frac {1} {q} \ in \ mathbb {q}\ backslash a \\ \ text {不是最小的元素} \ \ \}。据/annotation> 一种据/span>'据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>1据/span>∈据/span>问:据/span>\据/span>一种据/span>不是最小的元素据/span>}据/span>.据/span> 为了据S.pan class="katex"> X据/mi> 据据/mo> 0.据/mn> x < 0据/annotation> X据/span>据据/span>0.据/span>那据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> -据/mo> 1据/mn> )据/mo> x ^ { - 1} = - \左(\ ltvert x \ rvert ^ { - 1}右)据/annotation> X据/span>-据/span>1据/span>=据/span>-据/span>(据/span>|据/span>X据/span>|据/span>-据/span>1据/span>)据/span>.据/p> 对于Rational Cuts,所有上述算术运算都与Rational的算术运算一致。据/p>
对于实数据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> )据/mo> x = (A, \mathbb{Q} \反斜杠A)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>问:据/span>\据/span>一种据/span>)据/span>和据S.pan class="katex"> X据/mi> >据/mo> 0.据/mn> x > 0据/annotation> X据/span>>据/span>0.据/span>,乘法逆定义据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> (据/mo> 一种据/mi> '据/mo> 那据/mo> 问:据/mi> \据/mi> 一种据/mi> '据/mo> )据/mo> x^{-1} = (A', \mathbb{Q} \反斜杠A')据/annotation> X据/span>-据/span>1据/span>=据/span>(据/span>一种据/span>'据/span>那据/span>问:据/span>\据/span>一种据/span>'据/span>)据/span>, 在哪里据/p> 一种据/mi> '据/mo> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 1据/mn> 问:据/mi> ∈据/mo> 问:据/mi> \据/mi> 一种据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> a'= \ left \ {q \ in \ mathbb {q}:q \ leq 0 \ mbox {或} q> 0 \\ \ mbox {and} \ frac {1} {q} \ in \ mathbb {q}\ backslash a \\ \ text {不是最小的元素} \ \ \}。据/annotation> 一种据/span>'据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>1据/span>∈据/span>问:据/span>\据/span>一种据/span>不是最小的元素据/span>}据/span>.据/span> 为了据S.pan class="katex"> X据/mi> 据据/mo> 0.据/mn> x < 0据/annotation> X据/span>据据/span>0.据/span>那据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> -据/mo> 1据/mn> )据/mo> x ^ { - 1} = - \左(\ ltvert x \ rvert ^ { - 1}右)据/annotation> X据/span>-据/span>1据/span>=据/span>-据/span>(据/span>|据/span>X据/span>|据/span>-据/span>1据/span>)据/span>.据/p> 对于Rational Cuts,所有上述算术运算都与Rational的算术运算一致。据/p>
一种据/mi> '据/mo> =据/mo> {据/mo> 问:据/mi> ∈据/mo> 问:据/mi> :据/mo> 问:据/mi> ≤.据/mo> 0.据/mn> 或据/mtext> 问:据/mi> >据/mo> 0.据/mn> 和据/mtext> 1据/mn> 问:据/mi> ∈据/mo> 问:据/mi> \据/mi> 一种据/mi> 不是最小的元素据/mtext> }据/mo> .据/mi> a'= \ left \ {q \ in \ mathbb {q}:q \ leq 0 \ mbox {或} q> 0 \\ \ mbox {and} \ frac {1} {q} \ in \ mathbb {q}\ backslash a \\ \ text {不是最小的元素} \ \ \}。据/annotation> 一种据/span>'据/span>=据/span>{据/span>问:据/span>∈据/span>问:据/span>:据/span>问:据/span>≤.据/span>0.据/span>或据/span>问:据/span>>据/span>0.据/span>和据/span>问:据/span>1据/span>∈据/span>问:据/span>\据/span>一种据/span>不是最小的元素据/span>}据/span>.据/span>
为了据S.pan class="katex"> X据/mi> 据据/mo> 0.据/mn> x < 0据/annotation> X据/span>据据/span>0.据/span>那据S.pan class="katex"> X据/mi> -据/mo> 1据/mn> =据/mo> -据/mo> (据/mo> |据/mo> X据/mi> |据/mo> -据/mo> 1据/mn> )据/mo> x ^ { - 1} = - \左(\ ltvert x \ rvert ^ { - 1}右)据/annotation> X据/span>-据/span>1据/span>=据/span>-据/span>(据/span>|据/span>X据/span>|据/span>-据/span>1据/span>)据/span>.据/p> 对于Rational Cuts,所有上述算术运算都与Rational的算术运算一致。据/p>
对于Rational Cuts,所有上述算术运算都与Rational的算术运算一致。据/p>
给予实数据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> B.据/mi> )据/mo> x = (A, B)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>B.据/span>)据/span>和据S.pan class="katex"> y据/mi> =据/mo> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> 那据/mo> Y =(C,D),据/annotation> y据/span>=据/span>(据/span>C据/span>那据/span>D.据/span>)据/span>那据/span> X据/mi> X据/annotation> X据/span>小于或等于据S.pan class="katex"> y据/mi> 那据/mo> y,据/annotation> y据/span>那据/span>表示据S.pan class="katex"> X据/mi> ≤.据/mo> y据/mi> 那据/mo> X \ LEQ Y,据/annotation> X据/span>≤.据/span>y据/span>那据/span>如果据S.pan class="katex"> 一种据/mi> ⊆据/mo> C据/mi> .据/mi> 一个\ subseteq C。据/annotation> 一种据/span>⊆据/span>C据/span>.据/span>不等式是严格的据S.pan class="katex"> 一种据/mi> ⊂据/mo> C据/mi> .据/mi> a \ subset c.据/annotation> 一种据/span>⊂据/span>C据/span>.据/span>
实数的这种排序满足以下性质:据/p>
以上所有显示据S.pan class="katex"> R.据/mi> \ mathbb {r}据/annotation> R.据/span>是一个据S.trong>命令字段据/strong>.请注意,根据我们的定义,据S.pan class="katex"> ∞据/mi> \ infty据/annotation> ∞据/span>不是实数,因为据S.pan class="katex"> (据/mo> 问:据/mi> 那据/mo> ∅据/mi> )据/mo> (mathbb{Q}, \空集)据/annotation> (据/span>问:据/span>那据/span>∅据/span>)据/span>不是一个削减。此外,集合中有无数的元素据S.pan class="katex"> R.据/mi> \据/mi> 问:据/mi> \ mathbb {r} \ backslash \ mathbb {q}据/annotation> R.据/span>\据/span>问:据/span>,这是一组据一种href="//www.parkandroid.com/wiki/irrational-numbers/" class="wiki_link" title="不合理的数字" target="_blank">不合理的数字据/a>.据/p>
B.据/mi> ∈据/mo> R.据/mi> b \ in \ mathbb {r}据/annotation> B.据/span>∈据/span>R.据/span>是一个据S.trong>上界据/strong>对于一套据S.pan class="katex"> S.据/mi> ⊂据/mo> R.据/mi> s \ subset \ mathbb {r}据/annotation> S.据/span>⊂据/span>R.据/span>如果每个据S.pan class="katex"> S.据/mi> ∈据/mo> S.据/mi> \的年代据/annotation> S.据/span>∈据/span>S.据/span>满足据S.pan class="katex"> S.据/mi> ≤.据/mo> B.据/mi> s \ leq B据/annotation> S.据/span>≤.据/span>B.据/span>.如果据S.pan class="katex"> B.据/mi> B.据/annotation> B.据/span>是一个上限据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>和每个上限据S.pan class="katex"> B.据/mi> '据/mo> B'据/annotation> B.据/span>'据/span>满足据S.pan class="katex"> B.据/mi> '据/mo> ≥据/mo> B.据/mi> b'\ geq b据/annotation> B.据/span>'据/span>≥据/span>B.据/span>, 然后据S.pan class="katex"> B.据/mi> B.据/annotation> B.据/span>是据S.trong>最不上限据/strong>为了据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>.请注意,如果存在最小上限,则它是唯一的。据/p>
让据S.pan class="katex"> X据/mi> =据/mo> (据/mo> 一种据/mi> 那据/mo> B.据/mi> )据/mo> x =(a,b)据/annotation> X据/span>=据/span>(据/span>一种据/span>那据/span>B.据/span>)据/span>是剪裁。显示给定任何有理数据S.pan class="katex"> ϵ据/mi> >据/mo> 0.据/mn> \ε> 0据/annotation> ϵ据/span>>据/span>0.据/span>,存在有理数据S.pan class="katex"> 一种据/mi> ∈据/mo> 一种据/mi> 一个\在一个据/annotation> 一种据/span>∈据/span>一种据/span>和据S.pan class="katex"> B.据/mi> ∈据/mo> B.据/mi> b \在b据/annotation> B.据/span>∈据/span>B.据/span>这样据S.pan class="katex"> B.据/mi> -据/mo> 一种据/mi> 据据/mo> ϵ据/mi> b - a <\ epsilon据/annotation> B.据/span>-据/span>一种据/span>据据/span>ϵ据/span>.据/p> 根据定义,据S.pan class="katex"> 一种据/mi> 一种据/annotation> 一种据/span>和据S.pan class="katex"> B.据/mi> B.据/annotation> B.据/span>是不是空的,所以存在有理数据S.pan class="katex"> 一种据/mi> 0.据/mn> ∈据/mo> 一种据/mi> a_0 \在一个据/annotation> 一种据/span>0.据/span>∈据/span>一种据/span>和据S.pan class="katex"> B.据/mi> 0.据/mn> ∈据/mo> B.据/mi> b_0 \在b中据/annotation> B.据/span>0.据/span>∈据/span>B.据/span>.考虑理性数字的序列据S.pan class="katex"> (据/mo> 一种据/mi> 一世据/mi> 那据/mo> B.据/mi> 一世据/mi> )据/mo> 那据/mo> 一世据/mi> =据/mo> 1据/mn> 那据/mo> 2据/mn> 那据/mo> 3.据/mn> 那据/mo> ......据/mo> 那据/mo> (a_i, b_i), I = 1, 2, 3, ldots,据/annotation> (据/span>一种据/span>一世据/span>那据/span>B.据/span>一世据/span>)据/span>那据/span>一世据/span>=据/span>1据/span>那据/span>2据/span>那据/span>3.据/span>那据/span>......据/span>那据/span>被定义为据/p> 一种据/mi> 一世据/mi> =据/mo> {据/mo> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> 如果据/mtext> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> ∈据/mo> 一种据/mi> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> 否则据/mtext> B.据/mi> 一世据/mi> =据/mo> {据/mo> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> 如果据/mtext> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> ∈据/mo> B.据/mi> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 除此以外。据/mtext> ai = \开始{病例}\压裂{现代张{}+ b_张{}}{2}和{如果}\ \文本压裂{现代张{}+ b_张{}}{2}\中\ \现代张{}和{否则}\ \文本结束{病例}\ \ b_i = \开始{病例}\压裂{现代张{}+ b_张{}}{2}和{如果}\ \文本压裂{现代张{}+ b_张{}}{2}\ B \ \ b_张{}{否则& \文本。} \结束{病例}据/annotation> 一种据/span>一世据/span>=据/span>{据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>一种据/span>一世据/span>-据/span>1据/span>如果据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>∈据/span>一种据/span>否则据/span>B.据/span>一世据/span>=据/span>{据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>B.据/span>一世据/span>-据/span>1据/span>如果据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>∈据/span>B.据/span>除此以外。据/span> 注意据S.pan class="katex"> 一种据/mi> 一世据/mi> ∈据/mo> 一种据/mi> 那据/mo> B.据/mi> 一世据/mi> ∈据/mo> B.据/mi> a_i在A, b_i在B据/annotation> 一种据/span>一世据/span>∈据/span>一种据/span>那据/span>B.据/span>一世据/span>∈据/span>B.据/span>为了据S.pan class="katex"> 一世据/mi> =据/mo> 0.据/mn> 那据/mo> 1据/mn> 那据/mo> 2据/mn> 那据/mo> ......据/mo> i = 0,1,2,\ LDOTS据/annotation> 一世据/span>=据/span>0.据/span>那据/span>1据/span>那据/span>2据/span>那据/span>......据/span>和据S.pan class="katex"> B.据/mi> N.据/mi> -据/mo> 一种据/mi> N.据/mi> ≤.据/mo> 1据/mn> 2据/mn> N.据/mi> (据/mo> B.据/mi> 0.据/mn> -据/mo> 一种据/mi> 0.据/mn> )据/mo> b_n - a_n \ leq \ frac {1} {2 ^ {n}}(b_0-a_0)据/annotation> B.据/span>N.据/span>-据/span>一种据/span>N.据/span>≤.据/span>2据/span>N.据/span>1据/span>(据/span>B.据/span>0.据/span>-据/span>一种据/span>0.据/span>)据/span>.然后选择据S.pan class="katex"> N.据/mi> N.据/annotation> N.据/span>这样据S.pan class="katex"> 2据/mn> N.据/mi> >据/mo> B.据/mi> 0.据/mn> -据/mo> 一种据/mi> 0.据/mn> ϵ据/mi> 那据/mo> 2^{n} > \frac{b_0 - a_0}{\据/annotation> 2据/span>N.据/span>>据/span>ϵ据/span>B.据/span>0.据/span>-据/span>一种据/span>0.据/span>那据/span>理性的数字据S.pan class="katex"> 一种据/mi> N.据/mi> 那据/mo> B.据/mi> N.据/mi> a_n,b_n.据/annotation> 一种据/span>N.据/span>那据/span>B.据/span>N.据/span>满足的条件据S.pan class="katex"> 一种据/mi> N.据/mi> ∈据/mo> 一种据/mi> 那据/mo> B.据/mi> N.据/mi> ∈据/mo> B.据/mi> a_n在A, b_n在B据/annotation> 一种据/span>N.据/span>∈据/span>一种据/span>那据/span>B.据/span>N.据/span>∈据/span>B.据/span>和据S.pan class="katex"> B.据/mi> N.据/mi> -据/mo> 一种据/mi> N.据/mi> 据据/mo> ϵ据/mi> b_n - a_n <\ epsilon据/annotation> B.据/span>N.据/span>-据/span>一种据/span>N.据/span>据据/span>ϵ据/span>.据S.pan class="katex"> □据/mi> _\正方形据/annotation> □据/span>
根据定义,据S.pan class="katex"> 一种据/mi> 一种据/annotation> 一种据/span>和据S.pan class="katex"> B.据/mi> B.据/annotation> B.据/span>是不是空的,所以存在有理数据S.pan class="katex"> 一种据/mi> 0.据/mn> ∈据/mo> 一种据/mi> a_0 \在一个据/annotation> 一种据/span>0.据/span>∈据/span>一种据/span>和据S.pan class="katex"> B.据/mi> 0.据/mn> ∈据/mo> B.据/mi> b_0 \在b中据/annotation> B.据/span>0.据/span>∈据/span>B.据/span>.考虑理性数字的序列据S.pan class="katex"> (据/mo> 一种据/mi> 一世据/mi> 那据/mo> B.据/mi> 一世据/mi> )据/mo> 那据/mo> 一世据/mi> =据/mo> 1据/mn> 那据/mo> 2据/mn> 那据/mo> 3.据/mn> 那据/mo> ......据/mo> 那据/mo> (a_i, b_i), I = 1, 2, 3, ldots,据/annotation> (据/span>一种据/span>一世据/span>那据/span>B.据/span>一世据/span>)据/span>那据/span>一世据/span>=据/span>1据/span>那据/span>2据/span>那据/span>3.据/span>那据/span>......据/span>那据/span>被定义为据/p> 一种据/mi> 一世据/mi> =据/mo> {据/mo> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> 如果据/mtext> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> ∈据/mo> 一种据/mi> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> 否则据/mtext> B.据/mi> 一世据/mi> =据/mo> {据/mo> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> 如果据/mtext> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> ∈据/mo> B.据/mi> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 除此以外。据/mtext> ai = \开始{病例}\压裂{现代张{}+ b_张{}}{2}和{如果}\ \文本压裂{现代张{}+ b_张{}}{2}\中\ \现代张{}和{否则}\ \文本结束{病例}\ \ b_i = \开始{病例}\压裂{现代张{}+ b_张{}}{2}和{如果}\ \文本压裂{现代张{}+ b_张{}}{2}\ B \ \ b_张{}{否则& \文本。} \结束{病例}据/annotation> 一种据/span>一世据/span>=据/span>{据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>一种据/span>一世据/span>-据/span>1据/span>如果据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>∈据/span>一种据/span>否则据/span>B.据/span>一世据/span>=据/span>{据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>B.据/span>一世据/span>-据/span>1据/span>如果据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>∈据/span>B.据/span>除此以外。据/span> 注意据S.pan class="katex"> 一种据/mi> 一世据/mi> ∈据/mo> 一种据/mi> 那据/mo> B.据/mi> 一世据/mi> ∈据/mo> B.据/mi> a_i在A, b_i在B据/annotation> 一种据/span>一世据/span>∈据/span>一种据/span>那据/span>B.据/span>一世据/span>∈据/span>B.据/span>为了据S.pan class="katex"> 一世据/mi> =据/mo> 0.据/mn> 那据/mo> 1据/mn> 那据/mo> 2据/mn> 那据/mo> ......据/mo> i = 0,1,2,\ LDOTS据/annotation> 一世据/span>=据/span>0.据/span>那据/span>1据/span>那据/span>2据/span>那据/span>......据/span>和据S.pan class="katex"> B.据/mi> N.据/mi> -据/mo> 一种据/mi> N.据/mi> ≤.据/mo> 1据/mn> 2据/mn> N.据/mi> (据/mo> B.据/mi> 0.据/mn> -据/mo> 一种据/mi> 0.据/mn> )据/mo> b_n - a_n \ leq \ frac {1} {2 ^ {n}}(b_0-a_0)据/annotation> B.据/span>N.据/span>-据/span>一种据/span>N.据/span>≤.据/span>2据/span>N.据/span>1据/span>(据/span>B.据/span>0.据/span>-据/span>一种据/span>0.据/span>)据/span>.然后选择据S.pan class="katex"> N.据/mi> N.据/annotation> N.据/span>这样据S.pan class="katex"> 2据/mn> N.据/mi> >据/mo> B.据/mi> 0.据/mn> -据/mo> 一种据/mi> 0.据/mn> ϵ据/mi> 那据/mo> 2^{n} > \frac{b_0 - a_0}{\据/annotation> 2据/span>N.据/span>>据/span>ϵ据/span>B.据/span>0.据/span>-据/span>一种据/span>0.据/span>那据/span>理性的数字据S.pan class="katex"> 一种据/mi> N.据/mi> 那据/mo> B.据/mi> N.据/mi> a_n,b_n.据/annotation> 一种据/span>N.据/span>那据/span>B.据/span>N.据/span>满足的条件据S.pan class="katex"> 一种据/mi> N.据/mi> ∈据/mo> 一种据/mi> 那据/mo> B.据/mi> N.据/mi> ∈据/mo> B.据/mi> a_n在A, b_n在B据/annotation> 一种据/span>N.据/span>∈据/span>一种据/span>那据/span>B.据/span>N.据/span>∈据/span>B.据/span>和据S.pan class="katex"> B.据/mi> N.据/mi> -据/mo> 一种据/mi> N.据/mi> 据据/mo> ϵ据/mi> b_n - a_n <\ epsilon据/annotation> B.据/span>N.据/span>-据/span>一种据/span>N.据/span>据据/span>ϵ据/span>.据S.pan class="katex"> □据/mi> _\正方形据/annotation> □据/span>
一种据/mi> 一世据/mi> =据/mo> {据/mo> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> 如果据/mtext> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> ∈据/mo> 一种据/mi> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> 否则据/mtext> B.据/mi> 一世据/mi> =据/mo> {据/mo> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> 如果据/mtext> 一种据/mi> 一世据/mi> -据/mo> 1据/mn> +据/mo> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 2据/mn> ∈据/mo> B.据/mi> B.据/mi> 一世据/mi> -据/mo> 1据/mn> 除此以外。据/mtext> ai = \开始{病例}\压裂{现代张{}+ b_张{}}{2}和{如果}\ \文本压裂{现代张{}+ b_张{}}{2}\中\ \现代张{}和{否则}\ \文本结束{病例}\ \ b_i = \开始{病例}\压裂{现代张{}+ b_张{}}{2}和{如果}\ \文本压裂{现代张{}+ b_张{}}{2}\ B \ \ b_张{}{否则& \文本。} \结束{病例}据/annotation> 一种据/span>一世据/span>=据/span>{据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>一种据/span>一世据/span>-据/span>1据/span>如果据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>∈据/span>一种据/span>否则据/span>B.据/span>一世据/span>=据/span>{据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>B.据/span>一世据/span>-据/span>1据/span>如果据/span>2据/span>一种据/span>一世据/span>-据/span>1据/span>+据/span>B.据/span>一世据/span>-据/span>1据/span>∈据/span>B.据/span>除此以外。据/span>
注意据S.pan class="katex"> 一种据/mi> 一世据/mi> ∈据/mo> 一种据/mi> 那据/mo> B.据/mi> 一世据/mi> ∈据/mo> B.据/mi> a_i在A, b_i在B据/annotation> 一种据/span>一世据/span>∈据/span>一种据/span>那据/span>B.据/span>一世据/span>∈据/span>B.据/span>为了据S.pan class="katex"> 一世据/mi> =据/mo> 0.据/mn> 那据/mo> 1据/mn> 那据/mo> 2据/mn> 那据/mo> ......据/mo> i = 0,1,2,\ LDOTS据/annotation> 一世据/span>=据/span>0.据/span>那据/span>1据/span>那据/span>2据/span>那据/span>......据/span>和据S.pan class="katex"> B.据/mi> N.据/mi> -据/mo> 一种据/mi> N.据/mi> ≤.据/mo> 1据/mn> 2据/mn> N.据/mi> (据/mo> B.据/mi> 0.据/mn> -据/mo> 一种据/mi> 0.据/mn> )据/mo> b_n - a_n \ leq \ frac {1} {2 ^ {n}}(b_0-a_0)据/annotation> B.据/span>N.据/span>-据/span>一种据/span>N.据/span>≤.据/span>2据/span>N.据/span>1据/span>(据/span>B.据/span>0.据/span>-据/span>一种据/span>0.据/span>)据/span>.然后选择据S.pan class="katex"> N.据/mi> N.据/annotation> N.据/span>这样据S.pan class="katex"> 2据/mn> N.据/mi> >据/mo> B.据/mi> 0.据/mn> -据/mo> 一种据/mi> 0.据/mn> ϵ据/mi> 那据/mo> 2^{n} > \frac{b_0 - a_0}{\据/annotation> 2据/span>N.据/span>>据/span>ϵ据/span>B.据/span>0.据/span>-据/span>一种据/span>0.据/span>那据/span>理性的数字据S.pan class="katex"> 一种据/mi> N.据/mi> 那据/mo> B.据/mi> N.据/mi> a_n,b_n.据/annotation> 一种据/span>N.据/span>那据/span>B.据/span>N.据/span>满足的条件据S.pan class="katex"> 一种据/mi> N.据/mi> ∈据/mo> 一种据/mi> 那据/mo> B.据/mi> N.据/mi> ∈据/mo> B.据/mi> a_n在A, b_n在B据/annotation> 一种据/span>N.据/span>∈据/span>一种据/span>那据/span>B.据/span>N.据/span>∈据/span>B.据/span>和据S.pan class="katex"> B.据/mi> N.据/mi> -据/mo> 一种据/mi> N.据/mi> 据据/mo> ϵ据/mi> b_n - a_n <\ epsilon据/annotation> B.据/span>N.据/span>-据/span>一种据/span>N.据/span>据据/span>ϵ据/span>.据S.pan class="katex"> □据/mi> _\正方形据/annotation> □据/span>
表明对于任何非空的子集据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>的据S.pan class="katex"> R.据/mi> \ mathbb {r}据/annotation> R.据/span>有一个上限,存在最小的上限据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>.据/p> 一个非空的子集据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>的据S.pan class="katex"> R.据/mi> \ mathbb {r}据/annotation> R.据/span>是一套削减据S.pan class="katex"> {据/mo> (据/mo> C据/mi> 1据/mn> 那据/mo> D.据/mi> 1据/mn> )据/mo> 那据/mo> (据/mo> C据/mi> 2据/mn> 那据/mo> D.据/mi> 2据/mn> )据/mo> 那据/mo> (据/mo> C据/mi> 3.据/mn> 那据/mo> D.据/mi> 3.据/mn> )据/mo> 那据/mo> ......据/mo> }据/mo> \{(C_1,D_1), (C_2, D_2), (C_3, D_3), \ldots \}据/annotation> {据/span>(据/span>C据/span>1据/span>那据/span>D.据/span>1据/span>)据/span>那据/span>(据/span>C据/span>2据/span>那据/span>D.据/span>2据/span>)据/span>那据/span>(据/span>C据/span>3.据/span>那据/span>D.据/span>3.据/span>)据/span>那据/span>......据/span>}据/span>.让据/p> C据/mi> =据/mo> ∪据/mo> 一世据/mi> C据/mi> 一世据/mi> 那据/mo> D.据/mi> =据/mo> 问:据/mi> \据/mi> C据/mi> .据/mi> c = \ cup_i c_i,d = \ mathbb {q} \ backslash c.据/annotation> C据/span>=据/span>∪据/span>一世据/span>C据/span>一世据/span>那据/span>D.据/span>=据/span>问:据/span>\据/span>C据/span>.据/span> 然后据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (c,d)据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>是切还是上界据S.pan class="katex"> C据/mi> 一世据/mi> ⊆据/mo> C据/mi> c_i \ subseteq c据/annotation> C据/span>一世据/span>⊆据/span>C据/span>暗示据S.pan class="katex"> (据/mo> C据/mi> 一世据/mi> 那据/mo> D.据/mi> 一世据/mi> )据/mo> ≤.据/mo> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (C_i, D_i) \leq (C, D)据/annotation> (据/span>C据/span>一世据/span>那据/span>D.据/span>一世据/span>)据/span>≤.据/span>(据/span>C据/span>那据/span>D.据/span>)据/span>对所有人据S.pan class="katex"> 一世据/mi> 一世据/annotation> 一世据/span>.因为。的上界据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>存在,让我们据S.pan class="katex"> (据/mo> C据/mi> '据/mo> 那据/mo> D.据/mi> '据/mo> )据/mo> (光盘')据/annotation> (据/span>C据/span>'据/span>那据/span>D.据/span>'据/span>)据/span>是任何上限。然后通过定义上限,据S.pan class="katex"> C据/mi> '据/mo> C'据/annotation> C据/span>'据/span>包含据S.pan class="katex"> ∪据/mo> 一世据/mi> C据/mi> 一世据/mi> =据/mo> C据/mi> 那据/mo> \ cu_i C_i = C,据/annotation> ∪据/span>一世据/span>C据/span>一世据/span>=据/span>C据/span>那据/span>暗示据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> ≤.据/mo> (据/mo> C据/mi> '据/mo> 那据/mo> D.据/mi> '据/mo> )据/mo> (c,d)\ leq(c',d')据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>≤.据/span>(据/span>C据/span>'据/span>那据/span>D.据/span>'据/span>)据/span>.所以,据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (c,d)据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>是最不上限的据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>.据S.pan class="katex"> □据/mi> _\正方形据/annotation> □据/span>
表明对于任何非空的子集据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>的据S.pan class="katex"> R.据/mi> \ mathbb {r}据/annotation> R.据/span>有一个上限,存在最小的上限据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>.据/p>
一个非空的子集据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>的据S.pan class="katex"> R.据/mi> \ mathbb {r}据/annotation> R.据/span>是一套削减据S.pan class="katex"> {据/mo> (据/mo> C据/mi> 1据/mn> 那据/mo> D.据/mi> 1据/mn> )据/mo> 那据/mo> (据/mo> C据/mi> 2据/mn> 那据/mo> D.据/mi> 2据/mn> )据/mo> 那据/mo> (据/mo> C据/mi> 3.据/mn> 那据/mo> D.据/mi> 3.据/mn> )据/mo> 那据/mo> ......据/mo> }据/mo> \{(C_1,D_1), (C_2, D_2), (C_3, D_3), \ldots \}据/annotation> {据/span>(据/span>C据/span>1据/span>那据/span>D.据/span>1据/span>)据/span>那据/span>(据/span>C据/span>2据/span>那据/span>D.据/span>2据/span>)据/span>那据/span>(据/span>C据/span>3.据/span>那据/span>D.据/span>3.据/span>)据/span>那据/span>......据/span>}据/span>.让据/p> C据/mi> =据/mo> ∪据/mo> 一世据/mi> C据/mi> 一世据/mi> 那据/mo> D.据/mi> =据/mo> 问:据/mi> \据/mi> C据/mi> .据/mi> c = \ cup_i c_i,d = \ mathbb {q} \ backslash c.据/annotation> C据/span>=据/span>∪据/span>一世据/span>C据/span>一世据/span>那据/span>D.据/span>=据/span>问:据/span>\据/span>C据/span>.据/span> 然后据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (c,d)据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>是切还是上界据S.pan class="katex"> C据/mi> 一世据/mi> ⊆据/mo> C据/mi> c_i \ subseteq c据/annotation> C据/span>一世据/span>⊆据/span>C据/span>暗示据S.pan class="katex"> (据/mo> C据/mi> 一世据/mi> 那据/mo> D.据/mi> 一世据/mi> )据/mo> ≤.据/mo> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (C_i, D_i) \leq (C, D)据/annotation> (据/span>C据/span>一世据/span>那据/span>D.据/span>一世据/span>)据/span>≤.据/span>(据/span>C据/span>那据/span>D.据/span>)据/span>对所有人据S.pan class="katex"> 一世据/mi> 一世据/annotation> 一世据/span>.因为。的上界据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>存在,让我们据S.pan class="katex"> (据/mo> C据/mi> '据/mo> 那据/mo> D.据/mi> '据/mo> )据/mo> (光盘')据/annotation> (据/span>C据/span>'据/span>那据/span>D.据/span>'据/span>)据/span>是任何上限。然后通过定义上限,据S.pan class="katex"> C据/mi> '据/mo> C'据/annotation> C据/span>'据/span>包含据S.pan class="katex"> ∪据/mo> 一世据/mi> C据/mi> 一世据/mi> =据/mo> C据/mi> 那据/mo> \ cu_i C_i = C,据/annotation> ∪据/span>一世据/span>C据/span>一世据/span>=据/span>C据/span>那据/span>暗示据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> ≤.据/mo> (据/mo> C据/mi> '据/mo> 那据/mo> D.据/mi> '据/mo> )据/mo> (c,d)\ leq(c',d')据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>≤.据/span>(据/span>C据/span>'据/span>那据/span>D.据/span>'据/span>)据/span>.所以,据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (c,d)据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>是最不上限的据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>.据S.pan class="katex"> □据/mi> _\正方形据/annotation> □据/span>
C据/mi> =据/mo> ∪据/mo> 一世据/mi> C据/mi> 一世据/mi> 那据/mo> D.据/mi> =据/mo> 问:据/mi> \据/mi> C据/mi> .据/mi> c = \ cup_i c_i,d = \ mathbb {q} \ backslash c.据/annotation> C据/span>=据/span>∪据/span>一世据/span>C据/span>一世据/span>那据/span>D.据/span>=据/span>问:据/span>\据/span>C据/span>.据/span>
然后据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (c,d)据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>是切还是上界据S.pan class="katex"> C据/mi> 一世据/mi> ⊆据/mo> C据/mi> c_i \ subseteq c据/annotation> C据/span>一世据/span>⊆据/span>C据/span>暗示据S.pan class="katex"> (据/mo> C据/mi> 一世据/mi> 那据/mo> D.据/mi> 一世据/mi> )据/mo> ≤.据/mo> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (C_i, D_i) \leq (C, D)据/annotation> (据/span>C据/span>一世据/span>那据/span>D.据/span>一世据/span>)据/span>≤.据/span>(据/span>C据/span>那据/span>D.据/span>)据/span>对所有人据S.pan class="katex"> 一世据/mi> 一世据/annotation> 一世据/span>.因为。的上界据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>存在,让我们据S.pan class="katex"> (据/mo> C据/mi> '据/mo> 那据/mo> D.据/mi> '据/mo> )据/mo> (光盘')据/annotation> (据/span>C据/span>'据/span>那据/span>D.据/span>'据/span>)据/span>是任何上限。然后通过定义上限,据S.pan class="katex"> C据/mi> '据/mo> C'据/annotation> C据/span>'据/span>包含据S.pan class="katex"> ∪据/mo> 一世据/mi> C据/mi> 一世据/mi> =据/mo> C据/mi> 那据/mo> \ cu_i C_i = C,据/annotation> ∪据/span>一世据/span>C据/span>一世据/span>=据/span>C据/span>那据/span>暗示据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> ≤.据/mo> (据/mo> C据/mi> '据/mo> 那据/mo> D.据/mi> '据/mo> )据/mo> (c,d)\ leq(c',d')据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>≤.据/span>(据/span>C据/span>'据/span>那据/span>D.据/span>'据/span>)据/span>.所以,据S.pan class="katex"> (据/mo> C据/mi> 那据/mo> D.据/mi> )据/mo> (c,d)据/annotation> (据/span>C据/span>那据/span>D.据/span>)据/span>是最不上限的据S.pan class="katex"> S.据/mi> S.据/annotation> S.据/span>.据S.pan class="katex"> □据/mi> _\正方形据/annotation> □据/span>
认为据S.pan class="katex"> X据/mi> X据/annotation> X据/span>和据S.pan class="katex"> y据/mi> y据/annotation> y据/span>是真实的数字。证明以下陈述:据/p> 如果据S.pan class="katex"> X据/mi> ≤.据/mo> y据/mi> +据/mo> ϵ据/mi> X \ LEQ Y + \ epsilon据/annotation> X据/span>≤.据/span>y据/span>+据/span>ϵ据/span>对于每一个据S.pan class="katex"> ϵ据/mi> >据/mo> 0.据/mn> 那据/mo> \ε> 0,据/annotation> ϵ据/span>>据/span>0.据/span>那据/span>然后据S.pan class="katex"> X据/mi> ≤.据/mo> y据/mi> .据/mi> x \ leq y。据/annotation> X据/span>≤.据/span>y据/span>.据/span> 如果据S.pan class="katex"> |据/mo> X据/mi> -据/mo> y据/mi> |据/mo> 据据/mo> ϵ据/mi> 反转x - y反转< \据/annotation> |据/span>X据/span>-据/span>y据/span>|据/span>据据/span>ϵ据/span>对于每一个据S.pan class="katex"> ϵ据/mi> >据/mo> 0.据/mn> 那据/mo> \ epsilon> 0,据/annotation> ϵ据/span>>据/span>0.据/span>那据/span>然后据S.pan class="katex"> X据/mi> =据/mo> y据/mi> .据/mi> x = y。据/annotation> X据/span>=据/span>y据/span>.据/span> 通过订购原则,我们也有据S.pan class="katex"> X据/mi> ≤.据/mo> y据/mi> x \ leq y据/annotation> X据/span>≤.据/span>y据/span>或据S.pan class="katex"> X据/mi> >据/mo> y据/mi> x > y据/annotation> X据/span>>据/span>y据/span>.如果据S.pan class="katex"> X据/mi> >据/mo> y据/mi> x > y据/annotation> X据/span>>据/span>y据/span>然后让我们据S.pan class="katex"> ϵ据/mi> ε据/annotation> ϵ据/span>是区间内的任意数据S.pan class="katex"> 0.据/mn> 据据/mo> ϵ据/mi> 据据/mo> X据/mi> -据/mo> y据/mi> 0 < \ < x-y据/annotation> 0.据/span>据据/span>ϵ据/span>据据/span>X据/span>-据/span>y据/span>.这给了据S.pan class="katex-display"> ϵ据/mi> 据据/mo> X据/mi> -据/mo> y据/mi> ≤.据/mo> ϵ据/mi> 那据/mo> < x- y \leq \,据/annotation> ϵ据/span>据据/span>X据/span>-据/span>y据/span>≤.据/span>ϵ据/span>那据/span>矛盾。所以,据S.pan class="katex"> X据/mi> ≤.据/mo> y据/mi> x \ leq y据/annotation> X据/span>≤.据/span>y据/span>.据/p> 如果据S.pan class="katex"> X据/mi> ≠据/mi> y据/mi> x \ ne y据/annotation> X据/span>据/span>=据/span>y据/span>然后让我们据S.pan class="katex"> ϵ据/mi> ε据/annotation> ϵ据/span>是区间内的任意数据S.pan class="katex"> 0.据/mn> 据据/mo> ϵ据/mi> 据据/mo> |据/mo> X据/mi> -据/mo> y据/mi> |据/mo> 0 < \ - < lvert x - y \rvert据/annotation> 0.据/span>据据/span>ϵ据/span>据据/span>|据/span>X据/span>-据/span>y据/span>|据/span>.这给了据S.pan class="katex-display"> ϵ据/mi> 据据/mo> |据/mo> X据/mi> -据/mo> y据/mi> |据/mo> ≤.据/mo> ϵ据/mi> 那据/mo> \ epsilon <\ ltvert x - y \ rfter \ leq \ epsilon,据/annotation> ϵ据/span>据据/span>|据/span>X据/span>-据/span>y据/span>|据/span>≤.据/span>ϵ据/span>那据/span>矛盾。所以,据S.pan class="katex"> X据/mi> =据/mo> y据/mi> .据/mi> x = y。据/annotation> X据/span>=据/span>y据/span>.据/span> □据/mi> _\正方形据/annotation> □据/span>
认为据S.pan class="katex"> X据/mi> X据/annotation> X据/span>和据S.pan class="katex"> y据/mi> y据/annotation> y据/span>是真实的数字。证明以下陈述:据/p>
通过订购原则,我们也有据S.pan class="katex"> X据/mi> ≤.据/mo> y据/mi> x \ leq y据/annotation> X据/span>≤.据/span>y据/span>或据S.pan class="katex"> X据/mi> >据/mo> y据/mi> x > y据/annotation> X据/span>>据/span>y据/span>.如果据S.pan class="katex"> X据/mi> >据/mo> y据/mi> x > y据/annotation> X据/span>>据/span>y据/span>然后让我们据S.pan class="katex"> ϵ据/mi> ε据/annotation> ϵ据/span>是区间内的任意数据S.pan class="katex"> 0.据/mn> 据据/mo> ϵ据/mi> 据据/mo> X据/mi> -据/mo> y据/mi> 0 < \ < x-y据/annotation> 0.据/span>据据/span>ϵ据/span>据据/span>X据/span>-据/span>y据/span>.这给了据S.pan class="katex-display"> ϵ据/mi> 据据/mo> X据/mi> -据/mo> y据/mi> ≤.据/mo> ϵ据/mi> 那据/mo> < x- y \leq \,据/annotation> ϵ据/span>据据/span>X据/span>-据/span>y据/span>≤.据/span>ϵ据/span>那据/span>矛盾。所以,据S.pan class="katex"> X据/mi> ≤.据/mo> y据/mi> x \ leq y据/annotation> X据/span>≤.据/span>y据/span>.据/p>
如果据S.pan class="katex"> X据/mi> ≠据/mi> y据/mi> x \ ne y据/annotation> X据/span>据/span>=据/span>y据/span>然后让我们据S.pan class="katex"> ϵ据/mi> ε据/annotation> ϵ据/span>是区间内的任意数据S.pan class="katex"> 0.据/mn> 据据/mo> ϵ据/mi> 据据/mo> |据/mo> X据/mi> -据/mo> y据/mi> |据/mo> 0 < \ - < lvert x - y \rvert据/annotation> 0.据/span>据据/span>ϵ据/span>据据/span>|据/span>X据/span>-据/span>y据/span>|据/span>.这给了据S.pan class="katex-display"> ϵ据/mi> 据据/mo> |据/mo> X据/mi> -据/mo> y据/mi> |据/mo> ≤.据/mo> ϵ据/mi> 那据/mo> \ epsilon <\ ltvert x - y \ rfter \ leq \ epsilon,据/annotation> ϵ据/span>据据/span>|据/span>X据/span>-据/span>y据/span>|据/span>≤.据/span>ϵ据/span>那据/span>矛盾。所以,据S.pan class="katex"> X据/mi> =据/mo> y据/mi> .据/mi> x = y。据/annotation> X据/span>=据/span>y据/span>.据/span> □据/mi> _\正方形据/annotation> □据/span>
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