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.ver_mathstats said: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?
The statement m+n≠10 means that the sum of m and n is not equal to 10. This could be due to a variety of reasons, such as the values of m and n being different, or the sum of m and n being greater or less than 10.
Yes, for example, if m=6 and n=5, the sum of m and n is 11, which is not equal to 10. However, if m=3 and n=7, the sum of m and n is 10, which satisfies the equation m+n=10. This shows that a single example is not enough to prove that m+n≠10 in all cases.
To prove that m+n≠10, we need to show that it is not possible for the sum of any two numbers to equal 10. This can be done using mathematical proofs, such as by assuming that m+n=10 and showing that it leads to a contradiction, or by using counterexamples to show that m+n≠10 in certain cases.
Yes, it is possible for m+n to equal 10 in certain cases. For example, if m=5 and n=5, the sum of m and n is 10, which satisfies the equation m+n=10. However, as shown in the previous answers, m+n can also be not equal to 10 in other cases.
Understanding proofs about m+n≠10 is important because it allows us to logically and rigorously prove mathematical statements. It also helps us to develop critical thinking and problem-solving skills, which are essential for any scientific or mathematical research. Additionally, understanding proofs can help us to avoid making incorrect assumptions or conclusions based on a single example or observation.