For every a epsilon R, - ( - a) = a. Suggestion: What is meant by - (-a)? By A3,

Question

For every a epsilon R, - ( - a) = a. Suggestion: What is meant by - (-a)? By A3, the real number - (- a) is the unique real number with the property that when you add it to -a, you get 0. Now, what property do the real numbers -(-a) and a have in common? Both numbers have the property that when added to -a, the result is zero. What does the uniqueness part of the statement in A3 imply about -(-a) and a? For all a, b, c R, (a + b) + c = a + (b + c). There exists a unique number 0 epsilon R such that a + 0 = 0 + a = a for every a epsilon R. For all a epsilon R, there exists a unique number -a epsilon R such that a + (-a) = (-a) + a = 0. For all a, b epsilon R, a + b = b + a. For all a, b, c epsilon R, (a . b) . c = a middot (b . c). There exists a unique number 1 epsilon R such that a . 1 = 1 . a = a for every a epsilon R. For all nonzero a epsilon R. there exists a unique number a-1 epsilon R such that a . a-l = a-1. a = 1. For all a, b, c epsilon R, a . b = b . a. For all a, b, c R, a . (b + c) = a . b + a . c. NT1. 1 0. For all c R, exactly one of the following statements is true: 0

Explanation / Answer

Start with how -a is defined. It is "the additive inverse of a" (A3) - that is, it is the number that, when added to a, gives 0: a + -a = 0 Therefore -(-a) means the number that, when added to -a, gives 0. But applying the commutative property of addition, the equation above becomes -a + a = 0 Therefore the number that, when added to -a, gives 0 is a; or, -(-a) = a

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