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Homework - Mathematics 131A Homework 1 Due Friday April 8th...

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Unformatted text preview: Mathematics 131A: Homework 1 Due Friday, April 8th. Exercises 2.1, 2.4, 2.5, 2.9, 2.13, 2.14, 2.26, 2.29, 2.31, 2.33, 2.41, 2.53. True/False Statements 2.1, 2.10, 2.14, 2.17, 2.18, 2.19, 2.20. Below are the questions for those who do not have the book. Exercise 2.1 If A , B are any sets, show that A ∩ B and A \ B are disjoint and that A = ( A ∩ B ) ∪ ( A \ B ). Exercise 2.4 Let U be a universal set defined by U = { a,b,c,d,e,f, { a,b,c,d } , { a,d } , { d,e }} . Suppose that A = { a,b,e, { a,b,c,d } , { d,e }} , B = { a,b,c,f, { a,d } , { d,e }} , C = { b,c,f, { a,d }} . Compute the following: (1) A ∩ B , (3) A c ∩ B , (5) ( A \ B ) ∪ ( A \ C ), (7) A \ ( B ∪ C ), (2) A ∪ C , (4) ( A ∩ B ) ∪ ( A ∩ B c ), (6) ( A \ B ) ∩ ( A \ C ), (8) ( A \ B ) ∪ C . Exercise 2.5 Using truth tables, show that the statements: ( ¬ P ⇒ ( R ∧ ¬ R )) ⇔ P and ( P ⇒ Q ) ⇔ ( ¬ Q ⇒ ¬ P ) are both tautologies. Why are these tautologies significant in mathematics? Exercise 2.9 Show that f : X → Y has a well defined inverse function if and only if f is a bijection. Exercise 2.13 Recursively define a binary operation on N that is not associative. Exercise 2.14 Prove that 2 is prime. Exercise 2.26 Show that if a,b,c are natural numbers and a < b and b < c then a < c . Exercise 2.29 Prove that 1 3 + 2 3 + ··· + n 3 = (1 + ··· + n ) 2 for all natural numbers n . 1 Exercise 2.31 Prove that 1 2 + 4 2 + 7 2 + ··· + (3 n- 2) 2 = 1 2 n (6 n 2- 3 n- 1). Exercise 2.33 The extended principle of induction states that a list P ( m ) ,P ( m +1) ,... of propositions holds provided that (i) P ( m ) holds and (ii) P ( n + 1) holds whenever P ( n ) holds and n ≥ m . Prove that the axiom of induction implies the extended principle of induction. Exercise 2.41 Prove that n ! > 2 n if n ≥ 4. Exercise 2.53 Show that for any natural number K , there is an n large enough so that 2 n > K . True—False Questions. If the statement is true, prove the statement. If the statement is false, give a counterexample and then try to see if there is a way of modifying the statement so that the modified statement is true. Statement 2.1 Suppose that for each i ∈ N , A i is a set. Suppose further that every finite collection of the sets A i have non-empty intersection. Then the intersection of all of the A i ’s is also non-empty. (The statement below (2.10) is corrected from that given in the book.) Statement 2.10 Suppose that f : U → U and A,B ⊆ U . It is possible that A and B are disjoint but f ( A ) and f ( B ) are not disjoint. Let X and Y be two sets with equivalence relations = X and = Y respectively. Statement 2.14 A subset of ( X,Y ) may define a well-defined function with respect to = X and = Y but not define a well-defined function with respect to set equality....
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This note was uploaded on 09/23/2011 for the course MATH 131 taught by Professor Eskin during the Spring '10 term at UCLA.

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Homework - Mathematics 131A Homework 1 Due Friday April 8th...

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