Proof of the product of two odd integers

[acddklr06]

Hi Everyone!
I decided recently to start reading a book that acts as a transition to upper level mathematics. The last section of the chapter introduces you to the different proof techniques and mathematical facts to produce mathematical proofs. I think I understand everything, but I wanted to make sure by sharing my proof for a problem in the book. If anybody can chime in about if it is right or the like, please do so.

Homework Statement

The product of two odd integers is odd.

Homework Equations

N/A

The Attempt at a Solution

Let m and n be two odd integers. We will prove that if m and n are odd integers, then the product of m and n is odd. Since m and n are odd, there exists two integers, i and j, that are an element of Z such that m=2i+1 and n=2j+1. Substituting (2i+1) and (2j+1) into m*n, we produce (2i+1)(2j+1) =>4ij+2j+2i+1 => 2(2ij+j+i)+1, where (2ij+j+1) is an integer. Since (2ij+j+1) is an integer, there exists an integer k that is an element of Z such that (2ij+j+1)=k. By substituting k for (2ij+j+1), we produce 2k+1, which is the definition of an odd number. Therefore, the product of two odd integers is odd.

 

[Dick]

Looks fine to me. Though at some point you started substituting (2ij+j+1) for (2ij+j+i). I'm assuming that's just a typo.

 

[acddklr06]

Thank you for the quick response! It was definitely a typo. I just noticed it after reading your response.

 

[Mark44]

Also, don't use "implies" - ==> in place of "equals." Implication is used between two statements such that the first being true means that the one following will be true as well.
Equality is used to indicate that two expressions have the same value.
m*n = (2i + 1)(2j + 1) = 4ij + 2i + 2j + 1 = 2(2ij + i + j) + 1

 

[Unit]

The one suggestion I may have, which is purely stylistic and largely unmathematical, is that you reserve i and j for dealings with complex numbers. In this case I would've used p and q, so that m = 2p + 1 and n = 2q + 1. The reason I say that my suggestion is unmathematical is because once you define something to be something, it doesn't matter what readers may connote with it.