Utilizing Algebraic Properties in Basic Proofs

 

Objective 

 

Our goal is to learn how to implement basic algebra skills when we write our proofs. Specifically, we will focus on proving that integers are even or odd. Before we begin our lesson let us review the definitions of even and odd.

 

 

Even

 

An integer n is defined to be even when n=2k for some integer k.

 

 

 

Odd

 

An integer n is defined to be odd when n=2k +1 for some integer k.

 

Application

 

Now we will look at a basic proof in abstract mathematics.

 

If x is even then x² is even.

 

Notice that the statement above has an assumption and a conclusion. The assumption is that x is even while our conclusion is that x² is even.

 

Now here is the formal proof.

 

Proof:

 

Assume that x is even. [We wish to show that x² is even]

 

By definition of even x=2k for some integer k.

 

Therefore, x²= (2k) ²= 4k².

 

Moreover,  x² =4k²=2(2k²).

 

Since k is an integer then 2k² is an integer. Hence, we conclude that x² is even.

 

 

 

Now let us take a look at another proof.

 

If x is odd then x² is odd.

 

Proof:

 

Assume that x is odd. [We wish to show that x² is odd]

 

By definition of even x=2k+1 for some integer k.

 

Therefore, x²=(2k+1)²=4k² +4k+1.

 

Moreover, x²=4k² +4k+1=2(2k²+1)+ 1.

 

Since k is an integer, then 2k²+1 is an integer. Hence, x² is odd.