91Ó°ÊÓ

Which of the following sets are equal? a) \(\\{1,2,3\\}\) b) \(\\{3,2,1,3\\}\) c) \(\\{3,1,2,3\\}\) d) \(\\{1,2,2,3\\}\)

A manufacturer of 2000 automobile batteries is concerned about defective terminals and defective plates. If 1920 of her batteries have neither defect, 60 have defective plates, and 20 have both defects, how many batteries have defective terminals?

Joshua draws two ping-pong balls from a bowl of twenty ping-pong balls numbered 1 to 20 . Provide a sample space for this experiment if a) the first ball drawn is replaced before the second ball is drawn. b) the first ball drawn is not replaced before the second ball is drawn.

Ten ping-pong balls labeled 1 to 10 are placed in a box. Two of these balls are then drawn, in succession and without replacement, from the box. a) Find the sample space for this experiment. b) Find the probability that the label on the second ball drawn is smaller than the label on the first. c) Find the probability that the label on one ball is even while the label on the other is odd.

Prove each of the following results without using Venn diagrams or membership tables. (Assume a universe \(U\).) a) If \(A \subseteq B\) and \(C \subseteq D\), then \(A \cap C \subseteq B \cap D\) and \(A \cup C \subseteq B \cup D .\) b) \(A \subseteq B\) if and only if \(A \cap \bar{B}=\emptyset\). c) \(A \subseteq B\) if and only if \(\bar{A} \cup B=?\).

A professor has two dozen introductory textbooks on computer science and is concerned about their coverage of the topics \((A)\) compilers, \((B)\) data structures, and \((C)\) operating systems. The following data are the numbers of books that contain material on these topics: \(\begin{array}{lll}|A|=8 & |B|=13 \quad|C|=13 \\ |A \cap B|=5 & |A \cap C|=3 & |B \cap C|=6 \\ |A \cap B \cap C|=2 & & \end{array}\) (a) How many of the textbooks include material on exactly one of these topics? (b) How many do not deal with any of the topics? (c) How many have no material on compilers?

How many permutations of the 26 different letters of the alphabet contain (a) either the pattern "OUT" or the pattern "DIG"? (b) neither the pattern "MAN" nor the pattern "ANT"?

A set \(A\) has 128 subsets of even cardinality. (a) How many subsets of \(A\) have odd cardinality? (b) What is \(|A| ?\)

For \(A=\\{1,2,3,4,5,6,7\\}\), determine the number of a) subsets of \(A\) b) nonempty subsets of \(A\) c) proper subsets of \(A\) d) nonempty proper subsets of \(A\) e) subsets of \(A\) containing three elements f) subsets of \(A\) containing 1,2 g) subsets of \(A\) containing five elements, including 1,2 h) subsets of \(A\) with an even number of elements i) subsets of \(A\) with an odd number of elements

Using Venn diagrams, investigate the truth or falsity of each of the following, for sets \(A, B, C \subseteq U\). a) \(A \Delta(B \cap C)=(A \Delta B) \cap(A \Delta C)\) b) \(A-(B \cup C)=(A-B) \cap(A-C)\) c) \(A \Delta(B \triangle C)=(A \triangle B) \Delta C\)