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 B=
AB=
A-B=
B-C=
AC=
AC=
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 is a proper subset of every set." Is this correct or incorrect? Explain.
Problem 5: List the elements in the set { x Z | x2 - 7x + 5 = 0 }.
Problem 6: Let M = { y 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 (X) denote the power set of X. Find ({a, b, c, d}).
Problem 9: For each positive integer n, define the set An by An= {x 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 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 B) (C A)
b. (A ) A
c. (A B) (A B)
d. A (U - A)