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  3. 1 the operations of addition and multiplication of integers are...

Question: 1 the operations of addition and multiplication of integers are...

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1. The operations of addition and multiplication of integers are associative. That is, if m, n, p are integers, (m+n)+p-m+ (n+ p) and (m n).p- 2. The operations of addition and multiplication of integers are commutative. That is, if m, n are integers, m+n n+m and mn n m. n for all integers n. Furthermore, there is a unique integer, 1, such that n.1 1.n n for all integers n. 0 is called the additive identity and 1 is called the multiplicative 3. There is a unique integer, 0, such that n+0 0+n identity. 4. Every integer has an additive inverse. That is, if n is an integer, there is a unique integer m such that n + m = m + n = 0, We write-n for the unique additive inverse of n. 5. Multiplication distributes over addition. That is, if m, n,p are integers, m. (n +p)-mn+m.p and (m +n) p m p+n p.Exercise 1.1. Prove each of the following propositions about integers using only Properties 1-5 above and Proposition 1.1. (a) For all integers m and n, (m+ n)? m22mn+n2. (b) For all integers n, (-1): n =-n. Before proving this statement, explain carefully what (-1) .n means and what -n means. (c) For all integers m and n, (-m)n-(mn). (d) For all integers n, (-n) n?

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