A logic question...

[daon]

I am doing the following proof:

If \(\displaystyle m \leq n\) and \(\displaystyle n \leq m\) then \(\displaystyle m=n\).

I figured that I could do the following:

If \(\displaystyle m < n\) or \(\displaystyle m=n\) and \(\displaystyle n < m\) or \(\displaystyle n=m\) then \(\displaystyle m=n\).

So I believe I will have to break this into cases. Is my intuition correct? I Just don't know how to break a statement like this into cases:

If (A or B) and (C or D) then E

But B = D = E. :?:

Thanks,
Daon

 

[stapel]

I'd try contradiction: Suppose m != n. Since m < n and m != n, then m < n. Then m < n < m, so....

Eliz.

 

[pka]

Here the standard proof if at this point in your course you have the proposition.
If each of x and y is a real number then exactly one of these is true:
\(\displaystyle x < y\quad \underline \vee \quad x = y\quad \underline \vee \quad x > y\) .

\(\displaystyle \L
m \le n\quad \Rightarrow \quad m \not> n\)

\(\displaystyle \L
m \ge n\quad \Rightarrow \quad m \not< n\)

therefore there is only one possibility left \(\displaystyle \L
m = n\) .