Homework

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

Page1 / 12

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online