Syllabus | Calendar | Solutions
Solutions
PDF of odd-numbered solutions for Chapters 2 and 3
S 2.3 #100: This occurs when the sets A and B are disjoint, that is, they have no elements in common. For example, if A is the set of even natural numbers and set B is the set of odd natural numbers, then n(A ∩ B)=0, because there are no numbers that are both even and odd.
S 2.3 #104: Region I is defined by the set statement A ∩ B′.
Region II is defined by the set statement A ∩ B.
Region III is defined by the set statement A′ ∩ B.
Region IV is defined by the set statement A′ ∩ B′.
S 2.3 #116: A ∪ A′=U
S 2.3 #118: A ∩ emptyset=emptyset
S 2.4 #78: Region I is defined by the set statement A ∩ B′ ∩ C′.
Region II is defined by the set statement A ∩ B ∩ C′.
Region III is defined by the set statement A′ ∩ B ∩ C′.
Region IV is defined by the set statement A ∩ B′ ∩ C.
Region V is defined by the set statement A ∩ B ∩ C.
Region VI is defined by the set statement A′ ∩ B ∩ C.
Region VII is defined by the set statement A′ ∩ B′ ∩ C.
Region VIII is defined by the set statement (A ∪ B ∪ C)′.
S 2.5 #14: (If you got something different than what I got, please let me know!)
If set A is camping, set B is hiking, and set C is picnicking, then
Region I: 15
Region II: 45
Region III: 20
Region IV: 30
Region V: 95
Region VI: 50
Region VII: 35
Region VIII: 10
a) 290 had at least one of these features (i.e., A ∪ B ∪ C).
b) 95 had all three features (i.e., A ∩ B ∩ C).
c) 10 did not have any of these features (i.e., the complement of A ∪ B ∪ C).
d) 125 had exactly two of these features (i.e., Regions II, IV, and VI).
S 3.4 #54: To use De Morgan’s Law on the statement “we will go down to the marina and we will take out the sailboat,” let
q=textwewillgothemarina
and let
r=textwewilltakeoutthesailboat.
Then the statement “we will go down to the marina and we will take out the sailboat” can be written symbolically as
pwedgeq.
By one of De Morgan’s Laws, the negation of pwedgeq is
sim(pwedgeq)quad=quadsimp,veesimq,.
The contrapositive of the original conditional statement is, “If we do not go the marina or we do not take out the sailboat, then the sun is not shining.”
The converse of the original conditional statement is, “If we go down to the marina and we take the sailboat out, then the sun is shining.”
The inverse of the original conditional statement is, “If the sun is not shining, then we will not go down to the marina or we will not take out the sailboat.”
S 12.1 #2: The possible results of an experiment are called its outcomes.
S 12.2 #2: The probability of an event that cannot occur is 0.
S 12.2 #4: Every probability is a number between 0 and 1 inclusive.
S 12.2 #6: For any event A, P(A)+P(¬A)= 1.
S 12.2 #8: If the probability that an event occurs is 0.2, the probability that the event does not occur is 0.8 (because 1 minus 0.2 is 0.8).
S 12.4 #2: In an experiment, if an individual expects to have a loss in the long run, the expected value is negative.
S 12.4 #4: In an experiment, if an individual expects to break even in the long run, the expected value is $0.00.
S 12.6 #4: If it is impossible for two events A and B to occur simultaneously, then the events are considered to be mutually exclusive.
S 12.6 #6: For two events A and B, if the occurrence of either event has an effect on the probability of the occurrence of the other event, then the two events are considered to be dependent events.
S 12.6 #10: The formula for finding the probability of event A and event B is P(A ∧ B)=P(A) • P(B).
S 12.7 #4: If n(E1 ∧ E2)=5 and n(E1)=22, then P(E2 ∣ E1)=frac522.
S 12.8 #2: Any ordered arrangement of a given set of objects is called a permutation.
S 12.8 #4: The formula for the number of permutations of n distinct items is n!=n • (n−1) • (n−2) • ldots • 3 • 2 • 1.
S 12.8 #6: The number of permutations of n objects, where n1 items are identical, n2 of the items are identical, …, nr are identical is found by fracn!n1!n2!ldotsnr!.
S 12.9 #2: The symbol for the number of combinations when r items are selected from n distinct items is nCr.
S 12.9 #4: If we want to select r items from n items, and the order of the arrangement is not important, then combinations are used.
S 13.5 #12: Range is 0. Standard deviation is 0.
S 13.6 #18: Rectangular Distribution
S 13.6 #20: Bimodal Distribution
S 6.7 #2: y
S 6.7 #4: slope
S 6.7 #6: 2
S 6.7 #8: a) I, b) III
S 7.1 #2: solution
S 7.1 #4: consistent
S 7.1 #6: no
S 7.1 #8: infinite
S 7.1 #10: less
S 7.2 #2: no solution
S 6.8 #2: a) dashed, b) solid
S 7.5 #2: a dashed line
S 7.5 #4: ordered pairs