Utilizing Algebraic Properties in Basic Proofs
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.
An integer n is defined to be even when n=2k for some integer k.
An integer n is defined to be odd when n=2k +1 for some integer k.
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.
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.
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.