ass1 - ASSIGNMENT 1 - MATH235, FALL 2007 Submit by 16:00,...

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Unformatted text preview: ASSIGNMENT 1 - MATH235, FALL 2007 Submit by 16:00, Monday, September 17 (use the designated mailbox in Burnside Hall, 10th floor). 1. Calculate the following intersection and union of sets (provide short explanations, if not complete proofs). (1) Let N 0 be a natural number. What is N [-n, n]? What is N [-n, n]? n=1 n=1 (2) What is [n, n + 1]? What is (n, n + 2)? n=1 n=1 (3) Let An = {xn : x N}. What is An ? What is An ? n=1 n=1 2. Let A, B and C be sets. Prove or disprove: (1) (A \ B) \ C = A \ (B \ C); (2) (B \ A) (A \ B) = (A B) \ (A B). 3. Prove that A \ (iI Bi ) = iI (A \ Bi ). 4. Prove by induction that 13 + + n3 = (1 + + n)2 . You may use the formula for the r.h.s. we proved in class. 5. Let A = {1, 2, 3, 4} and B = {a, b, c}. (1) (2) (3) (4) (5) (6) (7) (8) Write 4 different surjective functions from A to B. Write 4 different injective functions from B to A. How many functions are there from A to B? How many surjective functions are there from A to B? How many injective functions are there from A to B? How many functions are there from B to A? How many surjective functions are there from B to A? How many injective functions are there from B to A? 6. Let f : A B be a function. a) Prove that f is bijective if and only if there exists a function g : B A such that f g = 1B and g f = 1A . b) Prove or disprove: if there exists a function g : B A such that f g = 1B then f is bijective. Notation: If a, b are real numbers we use the following notation: [a, b] = {x R|a x b}. [a, b) = {x R|a x < b}. (a, b] = {x R|a < x b}. (a, b) = {x R|a < x < b}. We also use [a, ) = {x R|a x}. (-, b] = {x R|x b}. (-, ) = R. If A1 , A2 , A3 , . . . are sets, we may write N Ai for A1 A2 AN and Ai for i{1,2,3,... } Ai . i=1 i=1 ...
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