w****o
发帖数: 367 | 1 以下是老师恢复我的问题的电子邮件，可我还是看不出来2种说法的不同。
而且我还可以证明，论题为真。可是一个简单的反例就证明论题为假。
怎么回事？谢谢大家！
"for every epsilon" is in the wrong place. It should say
"if (for every epsilon > 0, a <= b + epsilon) then (a<=b). "
As you point out, "for every epsilon > 0, (if a <= b + epsilon, then a<=b)
. " is not true.
> ________________________________________
> From:
> Sent: Monday, April 27, 2009 10:22 AM
> To:
>
> Dear Dr. J,
>
> Is the statement true or false?
>
> "for every epsilon > 0, if a <= b + epsilon, then a<=b. " | w****o
发帖数: 367 | 2 以下是我的证明：
"if (for every epsilon > 0, a <= b + epsilon) then (a<=b). "
"for every epsilon > 0, if a <= b + epsilon, then a<=b. "
Proof: suppose a>b. Let epsilon = (a-b)/2 > 0.
therefore, a <= b + epsilon = b + (a-b)/2 = (a+b)/2 = (2a-(a-b))/2 = a-
epsilon.
thus, a <= a - epsilon, i.e., epsilon <=0. a contradiction. Q.E.D.
我的证明有什么问题吗？反例命名说明这个命题是伪？哪里有问题？ | o****k
发帖数: 10 | 3 感觉你的证明是用第一个推出第一个。要证第二个，应该这样开始：
for any epsilon, suppose a <= b + epsilon but a>b……
这当然证不出。 | m*****n
发帖数: 3575 | 4 这个还是个很难的极限命题呢，因为你的反证是错误理解成for some epsilon，而这里
是every epsilon
for every epsilon>0
a=
<=>
a-b=
then a-b=<0
Suppose a>b holds, choose just one epsilon1=(a-b)/2>0
thus a-b>epsilon1 and a>b+epsilon
which contradicts the premise
for EVERY epsilon>0, a<=b+epsilon
Thus conclude
When the condition
(for every epsilon>0, a<=b+epsilon) holds
the condition
(a-b>0) cannot coexist (cannot be both true)
and it shall be a-b<=0 when the first condition is true
你的这个疑惑说明数学没有把逻辑学作为先修课，基础不扎实，呵呵。 | s*x
发帖数: 3328 | 5 楼主是F2转的F1吧......这个命题是对的，但是要这么理解：(就是教授回信里边的)
for all a/ for all b/ (for all epsilon>0/ a<=b+epsilon)-> a<=b
你后面的证明是对的，但是反例不是对应这个命题的。
这个命题的否命题是这个：（也就是你构造的反例应该满足的）
for some a/ for some b/ (for all epsilon>0/ a<=b+epsilon) and a>b
b)
【在 w****o 的大作中提到】
: 以下是老师恢复我的问题的电子邮件，可我还是看不出来2种说法的不同。
: 而且我还可以证明，论题为真。可是一个简单的反例就证明论题为假。
: 怎么回事？谢谢大家！
: "for every epsilon" is in the wrong place. It should say
: "if (for every epsilon > 0, a <= b + epsilon) then (a<=b). "
: As you point out, "for every epsilon > 0, (if a <= b + epsilon, then a<=b)
: . " is not true.
: > ________________________________________
: > From:
: > Sent: Monday, April 27, 2009 10:22 AM
| w****o
发帖数: 367 | 6 正解，多谢！
【在 m*****n 的大作中提到】
: 这个还是个很难的极限命题呢，因为你的反证是错误理解成for some epsilon，而这里
: 是every epsilon
: for every epsilon>0
: a=: <=>
: a-b=: then a-b=<0
: Suppose a>b holds, choose just one epsilon1=(a-b)/2>0
: thus a-b>epsilon1 and a>b+epsilon
: which contradicts the premise
| w****o
发帖数: 367 | 7 也是正解，多谢！
【在 s*x 的大作中提到】
: 楼主是F2转的F1吧......这个命题是对的，但是要这么理解：(就是教授回信里边的)
: for all a/ for all b/ (for all epsilon>0/ a<=b+epsilon)-> a<=b
: 你后面的证明是对的，但是反例不是对应这个命题的。
: 这个命题的否命题是这个：（也就是你构造的反例应该满足的）
: for some a/ for some b/ (for all epsilon>0/ a<=b+epsilon) and a>b
:
: b)
| w****o
发帖数: 367 | 8 我在最后总结一下：
1。"if (for every epsilon > 0, a <= b + epsilon) then (a<=b). " is true
The negation of it should be:
2。"(for every epsilon >0, a<=b+epsilon) and (a>b). "
我举的反例，不能证明了statement 2 is true，因为不能满足所有的epsilon.
3。"for every epsilon > 0, (if a <= b + epsilon, then a<=b) is wrong.
the negation of it should be:
4。There exists an epsilon > 0 such that (a <= b + epsilon and a>b)
我举的反例，恰好证明了statement 4 is true，所以，3 一定是伪。 | B********e
发帖数: 10014 | 9
【在 s*x 的大作中提到】
: 楼主是F2转的F1吧......这个命题是对的，但是要这么理解：(就是教授回信里边的)
: for all a/ for all b/ (for all epsilon>0/ a<=b+epsilon)-> a<=b
: 你后面的证明是对的，但是反例不是对应这个命题的。
: 这个命题的否命题是这个：（也就是你构造的反例应该满足的）
: for some a/ for some b/ (for all epsilon>0/ a<=b+epsilon) and a>b
:
: b)
| B********e
发帖数: 10014 | 10 omg! can't tolerate with it any more
it's simply a question of language
the statement is implictly in form of
(given a,b) if... then...
that's it! how can you guys get a whole lot of those stuff out?
if the question is understood in your way,
for all,for all, for all, then the conclusion is always,absolutely,
unconditionally right, because your assumption can not be true! how can
those three 'for all' happens at the same time?
another comrade said this is a very hard question, my god, if this is
【在 s*x 的大作中提到】
: 楼主是F2转的F1吧......这个命题是对的，但是要这么理解：(就是教授回信里边的)
: for all a/ for all b/ (for all epsilon>0/ a<=b+epsilon)-> a<=b
: 你后面的证明是对的，但是反例不是对应这个命题的。
: 这个命题的否命题是这个：（也就是你构造的反例应该满足的）
: for some a/ for some b/ (for all epsilon>0/ a<=b+epsilon) and a>b
:
: b)
| | | s*x
发帖数: 3328 | 11 if x+1=2 then x=1
that is more precisely: for all x/ x+1=2 -> x=1
where x is not a free variable, although the two are equivalent. if you don'
t agree with me, try to write a proof using any theory-prover, like PVS,
Isabelle/HOL, but I doubt you know any of them.
的)
【在 B********e 的大作中提到】
: omg! can't tolerate with it any more
: it's simply a question of language
: the statement is implictly in form of
: (given a,b) if... then...
: that's it! how can you guys get a whole lot of those stuff out?
: if the question is understood in your way,
: for all,for all, for all, then the conclusion is always,absolutely,
: unconditionally right, because your assumption can not be true! how can
: those three 'for all' happens at the same time?
: another comrade said this is a very hard question, my god, if this is
| s*x
发帖数: 3328 | 12 兄弟们，做人要谦虚，无知不可怕就怕不自知，can't tolerate一些人，no offense | B********e
发帖数: 10014 | 13 帅哥，你应该把数理逻辑耍到极致再加几个与或在前边，更显烧包
太仰慕你这种以把简单问题拽到无比复杂为己任的牛人了
don'
【在 s*x 的大作中提到】
: if x+1=2 then x=1
: that is more precisely: for all x/ x+1=2 -> x=1
: where x is not a free variable, although the two are equivalent. if you don'
: t agree with me, try to write a proof using any theory-prover, like PVS,
: Isabelle/HOL, but I doubt you know any of them.
:
: 的)
| B********e
发帖数: 10014 | 14 嘿嘿，不过俺得承认错读你的原文了，sorry
不是学逻辑的，可还是无知无畏的这么说：
你的一堆 for all a/ for all b/ if then...
逻辑上等价于 given a,b, if then
你说的新鲜名词俺真没听说过，还是记下了以后拽给别人听，谢了
【在 B********e 的大作中提到】
: omg! can't tolerate with it any more
: it's simply a question of language
: the statement is implictly in form of
: (given a,b) if... then...
: that's it! how can you guys get a whole lot of those stuff out?
: if the question is understood in your way,
: for all,for all, for all, then the conclusion is always,absolutely,
: unconditionally right, because your assumption can not be true! how can
: those three 'for all' happens at the same time?
: another comrade said this is a very hard question, my god, if this is
| m*****n
发帖数: 3575 | 15 all/some基本亚里十多德逻辑学
关于这个问题很难，我认为的难点是every epsilon这个概念本身的悖论问题，不多说
了。 | s*x
发帖数: 3328 | 16 该着你撞到我这里,说实话这些东西数学家是不研究的,只有两类人研究,一个是搞哲学
的,一个是搞计算机的,恰好我原来就搞过类似的形式化的东西(当然现在不搞这个了,觉
得很虚).这些玩意在计算机科学里边用处很大，比如写软件规范或者硬件设计上面，都
在用。我们学校计算机学院一个组的人在搞这个，专门给硬件系统作形式化验证的。
【在 B********e 的大作中提到】
: 嘿嘿，不过俺得承认错读你的原文了，sorry
: 不是学逻辑的，可还是无知无畏的这么说：
: 你的一堆 for all a/ for all b/ if then...
: 逻辑上等价于 given a,b, if then
: 你说的新鲜名词俺真没听说过，还是记下了以后拽给别人听，谢了
| B********e
发帖数: 10014 | 17 呵呵，大白话？
早上起来赵老师问你：吃早饭没？
你说：刚吃theory-prover，你呢？
赵老师说：我吃了PVS，加精的。你中午吃什么？
你说：我打算吃isabelle，听说补视力有效果，你呢？
赵老师说：紧！我还一直以为你吃HOL呢
的确够白，吼吼，有人说这个基本上很难，木想到对你都是
大白话，忒nb了，让我来敬仰你吧 |
|
