Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Need help proving this. Any help would be greatly appreciated. Thank you!

Let m, n $\in$ $\mathbb{Z}$. If m $\le$ n $\le$ m then m = n.

share|improve this question
What have you tried so far for this problem? –  NasuSama Feb 18 at 3:23
What is your definition of $\le$? Do you know it is antisymmetric? Presumably not, because this is proving it. We have to know what you have available to prove this. –  Ross Millikan Feb 18 at 3:25
@NasuSama I know that by definition this means either m<n and n<m or m=n, but I'm having trouble proving m=n. –  user127835 Feb 18 at 3:26
@RossMillikan If m>n it means m-n$ \in \mathbb{N}$. If m $\ge$ n it means m>n or m=n. –  user127835 Feb 18 at 3:27
Now, the statement seems reasonable. –  NasuSama Feb 18 at 3:28
add comment

4 Answers

You are given $(m \lt n) \vee (m=n)$, so assume $m \lt n$. Can you show $n \not \lt m$?

share|improve this answer
That makes it easy to prove $n \not \lt m$, doesn't it? –  Ross Millikan Feb 18 at 3:41
add comment

$m\leq n\leq m$ implies:

$1.$ $m$ is less than or equal to $n$

$2.$ $m$ is greater than or equal to $n$

Taking the intersection of the above two conditions, we get $m=n$.

share|improve this answer
add comment

By assuming $n\neq m$, you have either $n<m$ or $m<n$. Then assume one of it, say $n<m$, together with $n\geq m$. Can you get a contradiction?

share|improve this answer
add comment

Let's just start at the most complex side and do what comes naturally: \begin{align} & m \le n \;\land\; n \le m \\ \equiv & \qquad \text{"basic property of $\;\le\;$, twice -- to introduce $\;=\;$ as in our goal"} \\ & (m < n \lor m = n) \;\land\; (n < m \lor n = m) \\ \equiv & \qquad \text{"logic: factor out $\;m = n\;$ using distribution"} \\ & (m < n \;\land\; n < m) \lor m = n \\ \equiv & \qquad \text{"... -- to get rid of the left hand part"} \\ & \ldots & \end{align} Can you finish this calculation, given the rules you know about $\;<\;$?

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.