In this assignment, N will denote the set of positive integers, Z the set of all integers, Q the set of all rational numbers, and R the set of all real numbers. After any problem statement, feel free to hit the Enter key as often as you need to make space for your answer.

Problem 1:  Let A = {3,5,7}, B={2,3}, C = {1,2,3,4,7}.  Compute the following sets:

A Ç B =

A È B =

A - B =

B - C =

A Å C =

A ´ C =

Problem 2: Let D = {5, 2, {5,2}, {5, {2}}, {{a,b,c},{c,d,e}}} How many elements does D have?

Problem 3: A set E has 37 elements.  How many subsets does it have? (An answer correct to 5 significant digits will be acceptable.)

Problem 4: In 24x7 Section 1.4, the author states "The null set f is a proper subset of every set." Is this correct or incorrect? Explain.

Problem 5: List the elements in the set { x Î Z | x² - 7x + 5 = 0 }.

Problem 6: Let M = { y Î Q | 0 < y <= 1, and y can be written as a fraction with a denominator not exceeding 6. } List all the elements of M.  How many elements are there in M?

Problem 7: How many elements are there in the set {{{{{{{3}}}}}}} ?

Problem 8: Let Ã(X) denote the power set of X. Find Ã({a, b, c, d}).

Problem 9: For each positive integer n, define the set An by An= {x Î Z | n £ x £ 2n}

a. What is the union of all sets An?

b. What is the intersection of all sets An?

Problem 10: For every real number x, define Bx to be the open interval (-x,x).  Equivalently, Bx = {y Î R | |y| < x}.

a. What is the union of all sets Bx?

b. What is the intersection of all sets Bx?

Problem 11: Simplify each of the following algebraic expressions. All sets are assumed to be subsets of a universal set U.

a. (A È B) Ç (C È A)

b. (A È f) Ç A

c. (A È B) Ç (A Ç B)

d. A È (U - A)