**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 x ^{2} 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 x^{2} is even.

Now here is the formal proof.

**Proof:**

Assume that x is even. [We wish to show that x^{2 }is even]

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

Therefore, x^{2}= (2k)^{ 2}= 4k^{2}.

Moreover, x^{2} =4k^{2}=2(2k^{2}).

Since k is an integer then 2k^{2} is an integer. Hence, we conclude that x^{2} is even.

Now let us take a look at another proof.

*If x is odd then x ^{2} is odd.*

**Proof:**

Assume that x is odd. [We wish to show that x^{2 }is odd]

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

Therefore, x^{2}=(2k+1)^{2}=4k^{2} +4k+1.

Moreover, x^{2}=4k^{2} +4k+1=2(2k^{2}+1)+ 1.

Since k is an integer, then 2k^{2}+1 is an integer. Hence, x^{2} is odd.

**Subject :**Math**Topic :**Algebra 1, Algebra 2-
**Posted By :**Admin **Created on :**10-10-2010

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