Proofs Homework: Explain Why m+n≠10 Example Not Sufficient

[ver_mathstats]

Homework Statement

Suppose you are asked to prove that for all integers m and n, m+n≠10. You give the example m=1 and n=2. Why is this not sufficient?

Homework Equations

The Attempt at a Solution

I can't quite understand why it is not sufficient? Could someone please explain to me why is it insufficient?

 

[fresh_42]

Homework Statement

Suppose you are asked to prove that for all integers m and n, m+n≠10. You give the example m=1 and n=2. Why is this not sufficient?

Homework Equations

The Attempt at a Solution

I can't quite understand why it is not sufficient? Could someone please explain to me why is it insufficient?

What do you mean? $m+n\neq 10$ isn't true for some integers. It is sufficient to say $4+6=10$ in order to disprove it. As it cannot be proven, there cannot be a sufficient condition for a proof either.

An example is never sufficient to prove an all quantifier, only to disprove one.

 

[Orodruin]

Put slightly differently, you are asked to show that no integers n and m exist that satisfy n+m = 10. Then it is not enough to give a single example of integers that do not sum to 10 as another pair of integers could do that (and does).