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## The tutoring help you'll need in Math; from Geometry to Calculus

### Orlando, FL (32817)

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# Proof Technique

Proof techniques are fun. They are mathematical arguments of sufficiency. The feeling you get from arguing your point logically to completion is always exhilarating. Let us put our mathematical reasoning power to the test. Now since this is our first of many proof techniques. I will start with something elementary.

1. Prove the following:

(a) Given that the average of y and z is odd, then z is an odd integer when y is odd.

When doing proofs  it is best to do so in algebraic and generalized form. A way to represent all possible cases. Using only examples would be trivial, since we know that exceptions may exist.

Let n be some integer {-1, 0, 1, 2, ...}. An integer p is even if it can be represented in a form p = 2n. Trivial examples: 2*1=2, 2*2=4, 2*3=6 etc.

We will go a step further. An integer q is odd if it can be represented in a form q = 2n+1. Trivial examples: 2*1+1=3, 2*2+1=5, 2*3+1=7 etc.

Now we are armed and ready to begin the proofs.

1a) Let y be an odd integer, that is y = 2n+1. We will show that z must also be odd if the average of the two is also odd.

(y+z)/2 = 2n+1
y+z = 4n+2

But y can be expressed as 2n+1, since were told it is an odd integer.

2n+1+ z = 4n+2
z = 2n+1

We have shown that z = 2n +1 (which we stated is the form of an odd integer).

This is a general proof and can be shown for all odd integers, this is always true. Let look at three trivial examples.

(3+11)/2 = 7
(3+19)/2 = 11
(23+ 47)/2= 25

Try to prove the following: (a) Given that the average of y and z is even, then z is an odd integer when y is odd. Here are a few trivial examples to convince you to challenge the proof.

(3+13)/2=8
(3+113)/2= 116
(23+117)/2 = 70
(5+11115)/2 = 5560

There are many proof techniques. proof by Contrapositive, Contradiction, proof by Mathematical Induction, Biconditional proofs, Direct proofs, Element Chasing and Non-Constructive proofs. The above was a method of direct proof. More to come. Till then, Ciao!

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## Background Check Status for Kirk S.

Kirk S. passed a background check on 3/9/2011. You may run an updated background check on Kirk once you send an email.