Let a, b and c be real numbers. If a + b = a + c, then b = c. Suggestion : Use A
ID: 2943613 • Letter: L
Question
Let a, b and c be real numbers. If a + b = a + c, then b = c. Suggestion : Use Axiom A3, Axiom A1, and Axiom A2. 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 middot b) middot c = a middot (b middot c). There exists a unique number 1 R such that a middot 1 = 1 middot a = a for every a R. For all nonzero a R. there exists a unique number a-1 R such that a middot a-l = a-1 middot a = 1. For all a, b, c epsilon R, a middot b = b middot a. For all a, b, c R, a middot (b + c) = a middot b + a middot c. NT1. 1 0. For all c R, exactly one of the following statements is true: 0Explanation / Answer
a + b = a + c
Adding -a to both sides gives
LHS: -a + (a + b)
By A1, -a + (a + b) --> (-a + a) + b
By A3, (-a + a) + b --> 0 + b
By A2, 0 + b --> b
RHS: -a + (a + c)
By A1, -a + (a + c) --> (-a + a) + c
By A3, (-a + a) + c --> 0 + c
By A2, 0 + c --> c
Therefore, since LHS = RHS, b = c.
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