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By Armando Rojo

ISBN-10: 950025204X

ISBN-13: 9789500252041

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61 Theorem 6 is important because it can show that a mapping is invertible even though no simple formula for the inverse is known. For example, we can show (using calculus) that the function given by α(x) = 3 x + 2x is one-to-one and onto. But a simple formula for α−1 is not easy to write. 3 1. In each case, determine whether α is a well-defined mapping. Justify your answer. 2. In each case, state whether the mapping is onto, one-to-one, or bijective. Justify your answer. 62 3. Let A → αB → βC be mappings.

Then 1. α1A = α and 1Bα = α. 2. γ(βα) = (γβ)α. 3. If α and β are both one-to-one (both onto), the same is true of βα. Proof. (1) If a A, then α1A(a) = α [1A(a)] = α(a). Thus, α1A and α have the same action, that is, α1A = α. Similarly, 1Bα = α. (2) If a A: [γ(βα)](a) = γ[βα(a)] = γ[β(α(a))] = γβ[α(a)] = [(γβ)α](a). 57 (3) If α and β are one-to-one, suppose that βα(a) = βα(a1), where a, a1 A. Thus, β[α(a)] = β[α(a1)], so α(a) = α(a1) because β is one-to-one. But then a = a1 because α is one-to-one.

In each case, prove the result by contradiction and either prove the converse or give a counterexample. a. If n > 2 is a prime integer, then n is odd. b. If n + m = 25, where n and m are integers, one of n and m is greater than 12. 38 c. If a and b are positive numbers and a ≤ b, then (d) If m and n are integers and mn is even, then m is even or n is even. 4. Prove each implication by contradiction. a. If x and y are positive numbers, then b. If x is irrational and y is rational, then x + y is irrational.

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Algebra I 18ed by Armando Rojo

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