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

Question

For every a 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 R such that a+0= 0+a = a for every a R. For all a R, there exists a unique number -a R such that a + (-a) = (-a) + a = 0. For all a, b R, a + b = b + a. For all a,b,c R, (a . b) . c = a . (b . c). There exists a unique number 1 R such that a.1 = 1.a = a for every a R. For all nonzero a R, there exists a unique number a-1 R such that a .a-1 = a-1.a = 1. For all a, b R, a . b = b . a. For all a,b,c R, a.(b + c) = a.b + a.c. 1 0. For all a R, exactly one of the following statements is true: 0

Explanation / Answer

based on A3 we have (-(-a)) + (-a) = 0 now add a to both sides ((-(-a)) + (-a)) + a = a (-(-a)) + ((-a) + a) = a (A1) (-(-a)) + 0 = a (A3) (-(-a)) = a (A2)

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