A mathematics post on APOP is long overdue. In recognition of a current favorite topic of mine in Discrete Mathematics (proving or disproving propositions), I invite you to join me on a fun trip through evens and odds. My professor calls these baby proofs; they’re minor league mental preparation.

**Definitions**: For is if has the form , where .

For is if has the form , again where .

**Proposition**: The sum of two even integers is even.

**Pre-Proof Thoughts**: Testing with a few pairs of even integers (2 + 4, 10 + 20, 104 + 320), we find that the sums are even. However, examples are not sufficient to prove a proposition. We want to confirm that it holds for all even integers.

**Proof**: Let be even integers. Then and , where . Then we have:

, which is equivalent to

, which is equivalent to

, where by Closure of Addition

The result, , fulfills the definition for an even number. **QED**.

**Proposition**: The sum of two odd integers is even.

**Pre-Proof Thoughts**: We follow the same line of thought as before.

**Proof**: Let be odd numbers. Then and , where . Then we have:

, which is equivalent to

, which is equivalent to

, where by Closure of Addition

The result, , fulfills the definition for an even number. **QED**.

**Proposition**: The sum of an even integer and an odd integer is odd.

**Proof**: Let be even and be odd. Then and , where . Then we have:

, where by Closure of Addition

The result, , fulfills the definition for an odd number. **QED**.

**Proposition**: The product of an even integer and an odd integer is even.

**Proof**: Let be even and be odd. Then and , where . Then we have:

, which is equivalent to

, which is equivalent to

, which is equivalent to

, where by Closure of Multiplication,

which is equivalent to

, where by Closure of Addition

The result, , fulfills the definition for an odd number. **QED**.

**Proposition**: An integer cannot be both even and odd.

**Proof (by Contradiction)**: Suppose is both even and odd. Then and . where . Setting these equations equal to one another gives:

, which is equivalent to

There is no and such that . Thus, cannot be both even and odd, contradicting the proposition. **QED**.

Perhaps you disagree, but I for one find this proving business in post-Calculus undergraduate courses quite fun. Many aren’t as tidy as the even and odd proofs above, but once you go from proposition to end-of-proof symbol the satisfaction is unmatched.

*For proofs that draw illogical conclusions because of logical fallacies, Dr. Math FAQ has some examples for the infamous 1 = 2.*

*For information about writing in LaTeX in WordPress, see this article.*

I applaud your mathematical knowledge! Unfortunately, I understand none of this.

LikeLiked by 2 people

Haha, thanks, Candice! There’s still a ton of math out there that I don’t understand; hopefully by the end of my undergraduate years I know about some small fraction.

LikeLiked by 1 person