Contrapositive Proof: Ints $m$ & $n$ - Even/Odd Combinations

In summary, for a contrapositive proof, we need to show that if $m$ and $n$ are not both even or both odd, then $m+n$ is not even. This can be proven by assuming that $m$ is even and $n$ is odd, and showing that in this case, the negation of the premise holds, which is that $m+n$ is odd.
  • #1
tmt1
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For all integers $m$ and $n$, if $m+ n$ is even then $m$ and $n$ are both even or both odd.

For a contrapositive proof, I need to show that for all ints $m$ and $n$ if $m$ and $n$ and not both even and not both odd, then $ m + n $ is not even.

How do I go about doing this?
 
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  • #2
The negation of the conclusion is that exactly ONE of $m,n$ is odd.

In a proof of this type, you may assume $m$ is even, and $n$ is odd (or else we may "switch them").

Your mission, should you decide to accept it, is to prove that in this case, we have the negation of the premise:

that is, to show that $m+n$ is thus odd.
 

FAQ: Contrapositive Proof: Ints $m$ & $n$ - Even/Odd Combinations

What is a contrapositive proof?

A contrapositive proof is a type of mathematical proof that involves proving the statement "if A, then B" by instead proving the equivalent statement "if not B, then not A". This technique is often used to simplify complex proofs and is based on the logical fact that "if A implies B, then not B implies not A".

How is a contrapositive proof used to prove the even/odd combination of integers?

To prove that the sum of two integers, m and n, is even or odd, a contrapositive proof can be used by first assuming that the sum is not even or odd. This allows us to then prove that the individual integers, m and n, must also not be even or odd, respectively. By proving the contrapositive statement, we can then conclude that the original statement is true.

Can a contrapositive proof be used for any mathematical statement?

Yes, a contrapositive proof can be used for any mathematical statement that can be expressed as "if A, then B". However, it may not always be the most efficient or intuitive method of proof, and other techniques may be more appropriate depending on the specific statement being proved.

How does a contrapositive proof differ from a direct proof?

A direct proof involves proving a statement directly by using logical deductions and existing mathematical principles. A contrapositive proof, on the other hand, involves proving the statement indirectly by instead proving the equivalent contrapositive statement. While both methods can be used to prove the same statement, a contrapositive proof may be more useful in certain situations, such as when the original statement is difficult to prove directly.

Are there any drawbacks to using a contrapositive proof?

One potential drawback of using a contrapositive proof is that it may not provide as much insight or understanding of the statement being proved compared to a direct proof. It also requires the use of logical equivalencies and may be more difficult for some individuals to understand compared to other proof techniques. Additionally, it may not always be the most efficient method of proof and other techniques may be more suitable.

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