klarfeld-sec-5-1 - Jennifer Klarfeld Sec 5.1 Edition 8 1...

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Jennifer Klarfeld – Sec. 5.1 – Edition 8 1. Prove Theorem 5.1.1: Show that the relation is reflexive, symmetric, and transitive on the class of all sets. Let A , B and C be sets in the class of all sets, and let be the relation “equivalence” of sets. (i) For all sets A in the class of all sets, A ≈ A since there exists a one-to-one function from A A , namely I A : A A defined by I A ( x ) = x for all x A . Therefore, is reflexive. (ii) Suppose A ≈B. Then there exists f : A B that is one-to-one and onto B. By Corollary 4.4.3, f 1 : B A is also one-to-one and onto A, so B≈ A. Therefore, is symmetric. (iii) Suppose A ≈B and B≈C . Then there exists f : A B that is one-to-one and onto B and g : B C that is one-to-one and onto C. By Theorem 4.4.1, g f : A C is a one-to-one correspondence. So, A ≈C . Therefore, is transitive. 1
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Jennifer Klarfeld – Sec. 5.1 – Edition 8 4. Complete the proof that any two open intervals ( a,b ) and ( c ,d ) are equivalent by showing that f ( x ) = d c b a ( x a ) + c maps one-to-one and onto ( c ,d ) .
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