MGFReview # 1----------------------

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Decide whether the following is a statement or is not a statement.1) This test is too hard. 1)

Answer: Not a statement

2) 4 + 2 = 7 2)Answer: Statement

Decide whether the statement is compound.3) The longest book I ever read was "War and Peace." 3)

Answer: Not compound

4) 2 is rational and 8 is irrational. 4)Answer: Compound

Let p represent a true statement, and let q and r represent false statements. Find the truth value of the given compoundstatement.

5) q ~r 5)Answer: True

6) ~[(~p q) r] 6)Answer: True

7) ~(p q) (r ~q) 7)Answer: True

8) ~[~p (~q p)] 8)Answer: True

1

Translate the symbolic compound statement into words.9) Let p represent the statement "Her name is Lisa" and let q represent the statement "She

lives in Chicago."p ~q

9)

Answer: Her name is Lisa or she does not live in Chicago.

10) Let p represent the statement "Students are males" and let q represent the statement"Teachers are males."~(p ~q)

10)

Answer: It is not the case that students are males or teachers are not males.

Let p represent the statement "Jim plays football" and let q represent the statement "Michael plays basketball." Convertthe compound statement into symbols.

11) Jim does not play football and Michael does not play basketball. 11)Answer: ~p ~q

12) Jim does not play football and Michael plays basketball. 12)Answer: ~p q

13) Jim does not play football or Michael plays basketball. 13)Answer: ~p q

Let p represent the statement 7 < 8, let q represent the statement 2 < 5 < 6, and let r represent the statement 3 < 2. Find thetruth value of the given compound statement.

14) (~r ~q) (~r q) 14)Answer: True

15) (q ~p) ~q 15)Answer: True

Write a negation for the statement.16) x < 64 16)

Answer: x 64

17) My brother is asleep. 17)Answer: My brother is not asleep.

2

18) That athlete wants to be a musician. 18)Answer: That athlete does not want to be a musician.

Construct a truth table for the compound statement.19) r ~(c s) 19)

Answer:r c s r ~(c s)

T T T T T T F T T F T T T F F T F T T F F T F T F F T T F F F T

20) ~p ~s 20)Answer:

p s (~p ~s) T T F T F FF T FF F T

Use one of De Morgan's laws to write the negation of the statement.21) I was a day late and a dollar short. 21)

Answer: I was not a day late or not a dollar short.

22) Roger or Emil will attend the game. 22)Answer: Roger will not attend the game and Emil will not attend the game.

23) Cats are lazy or dogs aren't friendly. 23)Answer: Cats aren't lazy and dogs are friendly.

Determine whether the statement is true or false.24) If p is true, then ~p (q r) is false. 24)

Answer: False

3

Let p represent "the puppy behaves well," let q represent "the puppy's owners are happy," and let r represent "the puppyis trained." Express the compound statement in words.

25) ~r ~q 25)Answer: If the puppy is not trained, then his owners are not happy.

26) ~(p q) 26)Answer: It is not the case that if the puppy behaves well, then his owners are happy.

27) p r 27)Answer: If the puppy behaves well, then the puppy is trained.

Let p represent "I eat too much," let q represent "I exercise," and let r represent "the food is good." Write the compoundstatement in symbols.

28) If the food is not good, then I don't eat too much. 28)Answer: ~r ~p

29) If I exercise, then I don't eat too much. 29)Answer: q ~p

30) If I eat too much, then I exercise. 30)Answer: p q

Find the truth value of the statement. Assume that p and q are false, and r is true.31) r (~p q) 31)

Answer: True

32) (~p ~q) (p ~r) 32)Answer: False

33) ~p (q r) 33)Answer: True

4

Construct a truth table for the statement.34) (q ~r) (q ~r) 34)

Answer:q r (q ~r) (q ~r) T T T T F T F T F F F F

Identify the statement as true or false.35) 7 x 2 = 18 if and only if 4 + 9 = 13. 35)

Answer: False

36) 4 + 7 13 if and only if 8 × 5 45. 36)Answer: True

Construct a truth table for the statement.37) ~p (~p t) 37)

Answer:p t ~p (~p t) T T T T F T F T T F F F

38) (p ~q) (p q) 38)Answer:

p q ( p ~q) (p q) T T F F F T T F T T F F F T F F F T F F F T F T

(2) (1) (4) (3)

Write the statement as an equivalent statement that does not use the if . . . then connective. Remember that p q isequivalent to ~p q.

39) If the sun comes out Tuesday, the daisies will open. 39)Answer: The sun does not come out Tuesday or the daisies will open.

5

Write the negation of the statement. Remember that the negation of p q is p ~q.40) If it is raining, you take your umbrella. 40)

Answer: It is raining and you do not take your umbrella.

For the given direct statement, write the indicated related statement (converse, inverse, or contrapositive).41) If I pass, then I'll party. (contrapositive) 41)

Answer: If I don't party, then I didn't pass.

42) If I were young, I would be happy. (converse) 42)Answer: If I were happy, I would be young.

Decide whether the argument is valid or invalid using the truth table.43) If you read, then you will have a high score.

You do not read.You will not have a high score.

43)

Answer: Invalid; fallacy of the inverse

44) If the bough breaks, then the cradle will fall.The bough breaks.The cradle will fall.

44)

Answer: Valid; modus ponens

45) If I'm hungry, then I will eat.I'm not hungry.I will not eat.

45)

Answer: Invalid; fallacy of the inverse

6

Find the number of subsets of the set.46) {-15, 0, 15, 16, 17} 46)

Answer: 32

47) {mom, dad, son, daughter} 47)Answer: 16

Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicatedset, using set braces.

48) (A B)' 48)Answer: {r, t, u, v, w, x, z}

49) A' B 49)Answer: {q, r, s, t, v, x, y, z}

50) A (B C) 50)Answer: {q, s, u, w, y, z}

51) B' (A C') 51)Answer: {r, t, u, w}

The lists below show five agricultural crops in Alabama, Arkansas, and Louisiana.

Alabama Arkansas Louisianasoybeans (s) soybeans (s) soybeans (s)peanuts (p) rice (r) sugarcane (n)corn (c) cotton (t) rice (r)hay (h) hay (h) corn (c)wheat (w) wheat (w) cotton (t)

Let U be the smallest possible set that includes all of the crops listed; and let A, K, and L be the sets of five crops inAlabama, Arkansas, and Louisiana, respectively. Find the indicated set.

52) A K 52)Answer: {h, s, w}

7

Shade the Venn diagram to represent the set.53) A' B' 53)

Answer:

54) (A B C')' 54)

Answer:

8

55) (A B C')' 55)

Answer:

Solve the problem.56) Jose is applying to college. He receives information on 8 different colleges. He will apply

to all of those he likes. He may like none of them, all of them, or any combination of them.How many possibilities are there for the set of colleges that he applies to?

56)

Answer: 256

Use the union rule to answer the question.57) If n(A) = 5, n(B) = 11, and n(A B) = 3; what is n(A B)? 57)

Answer: 13

58) If n(B) = 48, n(A B) = 9, and n(A B) = 84; what is n(A)? 58)Answer: 45

59) If n(A) = 25, n(A B) = 73, and n(A B) = 21; what is n(B)? 59)Answer: 69

9

60) If n(A) = 16, n(B) = 45, and n(A B) = 53; what is n(A B)? 60)Answer: 8

Find the probability of the given event.61) A single fair die is rolled. The number on the die is prime. 61)

Answer: 12

62) A single fair die is rolled. The number on the die is a 3 or a 4. 62)

Answer: 13

Find the probability.63) When a single card is drawn from a well-shuffled 52-card deck, find the probability of

getting a black card.63)

Answer: 12

64) When a single card is drawn from a well-shuffled 52-card deck, find the probability ofgetting a black 7.

64)

Answer: 126

65) A bag contains 9 red marbles, 6 blue marbles, and 2 green marbles. What is the probabilitythat a randomly selected marble is blue?

65)

Answer: 617

66) A bag contains 8 red marbles, 4 blue marbles, and 1 green marble. What is the probabilitythat a randomly selected marble is not blue?

66)

Answer: 913

10

67) A bag contains 19 balls numbered 1 through 19. What is the probability that a randomlyselected ball has an even number?

67)

Answer: 919

68) A lottery game has balls numbered 1 through 15. What is the probability that a randomlyselected ball is an even numbered ball or a 6?

68)

Answer: 715

Find the probability of the given event.69) A single fair die is rolled. The number on the die is not 6. 69)

Answer: 56

Find the probability.70) Each digit from the number 9,575,225 is written on a different card. If one of these cards is

selected at random, what is the probability of drawing a card that shows 9, 5, or 7?70)

Answer: 57

71) A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawingan ace or a 6?

71)

Answer: 213

Use a Venn diagram to find the indicated probability.72) If P(A B) = 0.65, P(A) = 0.32, and P(B) = 0.48, find P(A B). 72)

Answer: 0.15

11

Find the indicated probability.73) Find the probability that either a 1 or a 5 is obtained when a fair die is rolled. 73)

Answer: 13

74) Find the probability that the sum is no more than 3 when two fair dice are rolled. 74)

Answer: 112

Find the odds.75) Find the odds in favor of drawing a red marble when a marble is selected at random from

a bag containing 2 yellow, 5 red, and 6 green marbles.75)

Answer: 5 to 8

Find the indicated probability.76) Find the probability that the sum is either 9 or at most 3 when two fair dice are rolled. 76)

Answer: 736

Use a Venn diagram to find the indicated probability.77) If P(A B) = 0.61, P(A) = 0.35, and P(B) = 0.46, find P(A B). 77)

Answer: 0.2

78) If P(A B) = 0.21, P(A) = 0.38, and P(B) = 0.46, find P(A B). 78)Answer: 0.63

79) If P(A B) = 0.62, P(A) = 0.35, and P(A B) = 0.16, find P(B). 79)Answer: 0.43

Find the odds.80) Find the odds in favor of spinning an A on the spinner pictured below. (The sectors are of

equal size.)80)

Answer: 3 to 5

12

81) Find the odds in favor of rolling an odd number when a fair die is rolled. 81)Answer: 1 to 1

82) Find the odds in favor of rolling a number less than 3 when a fair die is rolled. 82)Answer: 1 to 2

83) Find the odds in favor of getting a sum of 9 when two fair dice are rolled. 83)Answer: 1 to 8

84) Find the odds in favor of getting a sum of 5 or 8 when two fair dice are rolled. 84)Answer: 1 to 3

85) Find the odds against correctly guessing the answer to a multiple choice question with 5possible answers.

85)

Answer: 4 : 1

13