Example for Proofs

  • #1
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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?
 

Answers and Replies

  • #2
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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.
 
  • #3
Orodruin
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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).
 

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