Direct Mathematical Proofs

In this section, we will look into how mathematical proofs are constructed and understand this with a few simple examples.

The simplest way to establish a mathematical truth is through a direct proof that shows the definitions of the terms led through a sequence of deductions that lead to the conclusion we wish to prove.

Let's look at a simple example and construct our own proof showing that the product of an even and an odd integer is itself an even number.

Example – Products of Even and Odd Integers

Let x be an even integer. This means x is a multiple of 2, so there exists an integer n where we have the following:

x = 2n

Let y be an odd integer. This means y is not a multiple of 2, which means when we divide it by 2, we will have a remainder of 1, which means there is an integer m such that we have the following:

y = 2m + 1

If we multiply them together, we find the following:

xy = (2n)(2m + 1)

xy = 4nm + 2n

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