# Associative and Commutative Rule

Prove that addition modulo n, written +n is:

1) associative

2)comutative

there are two ways to prove these properties. each way requires a definition or two:

1) for n≥2, 0≤a, b≤n+1

a+n(written as a power in a corner downside, but dont know how to put it tho) b={condition 1 - a+b if a+b<n;condition 2 - a+n-n if a+b>or=n}

2) writing an(n is written as a power at the corner downside) for a mod n and (a+b)n(little n at the corner)=a+n(little n at the corner downside)b, then

(p+nq)≡(Pn+qn)n (again n is written as a power at the corner but down the side not on the top)

Do the proofs using both methods. which is more algebraic?

I need you to go through the proof, step by step with me in this question, and how to apply these condition to the two methods of proving.

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#### Solution Preview

I'm going to type up the mathematical part of the answer in Microsoft Word and attach it separately. The basic vitally important definitions are here.

Please let me know if it comes through OK because I'm new to this system and want to make sure everything works out all right -- there's some way here you can drop a message, I think (if nothing else works, try tos end a problem with zero credits and put my name on it)

OK, two very important basics.

Commutatitive and Associative:

To COMMUTE means to drive back and forth -- for example, your dad or your neighbour commutes to work, ie he/she drives in and out every day. So the commutative law has to do with going back and forth, reversing direction.

In basic arithmetic, we use this all the time without thinking about it:

5 + 9 = 9 + 5 , a + b = b + a (commutative law of addition)

5 x 9 = 9 x 5 , ab = ba (commutative law of multiplication)

Note that some things are NOT commutative -- you can't change order in subtraction or in division, right?

NOT correct: 9 - 5 X=X 5 - 9 (using X=X to mean NOT equal

2/4 X=X 4/2

Also if you have learned matrices, matrix multiplication is NOT commutative, you can;t change the order. It's the fact that some things work and some things don't that make it important to study when the property works and when it doesn't.

To ASSOCIATE means to get together in groups. You associate with your friends. Your co-workers my also be called your associates. So the ...