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Wilfrid Laurier COURSE MA 215, Summer 2014




Question;MA215 Set Theory;Midterm Test I (Fall, 2013);SOLUTIONS;1. (15 marks) A sequence an is de?ned recursively by a1 = 1, a2 = 3 and an = 2an1 an2 for n 3. Use the strong;version of mathematical induction to prove that an = 2n 1 for all n 2 P.;2. (10 marks) Prove De Morgan?s law: (A [ B)c = Ac \ Bc.;3.;(15 marks) Let H = f2;m;m 2;Zg;Q;+;the set;of positive rational;a;numbers. A relation R is de?ned;on Q+ by aRb if;2 H.;b;(a) Show that R is an equivalence relation on Q+.;1;(b) Describe the elements in the equivalence;classes;and;[6].;8;a;6;1;4. (20;marks) Let A = f1, 2g and B = fa, b, cg.;(b) Find the number of functions from A to B.;5. (15 marks) Let R be a relation;on A. The symmetric closure of R, P(R), is a relation on A with the properties;(1) it is symmetric, (2) R P(R) and (3);if S is a symmetric relation on A with R S, then P(R) S.;T;(a) Prove that S2M S is a symmetric;relation on A containing R, where;M is the collection of all symmetric relations on A containing R.;T;S;(b);Prove that P(R) =;S2MS.;T;2M;S2M;S;by (a) above, it follows from Property (3);of the de?nition of;P;(R) that;P(R);T;T;T;S2MS.;A2M;T;\;6. (10;marks) Consider functions f: A ! B and g: B ! C.;(a) Prove that if g f is;one-to-one, then f is one-to-one.;(b) Prove that if g f is onto, then g is onto.;2;7.(15 mark) Consider the set of;integers Z together with the arithmetic operations +, and. Assume that + and;have the usual properties such as commutativity, associativity, cancellations;and distributions. Let A = Z (Z n f0g) and de?ne a relation on A by (a, b) (c;d) if and only if ad = bc.;(a);Verify that the relation is an equivalence;relation on A = Z (Znf0g).;(b) Let Q;= (Z (Z n f0g))=, where;is the equivalence relation in part;(a). Prove that the operation given by [(a, b)] [(c, d)] = [(ad;bc)], where c 6= 0, is well-de?ned.;d;c.;3


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