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Math 415 --- Assignment 1 Rough Answers 1.2.1 The expressions (1),(4),(5),(6) are statements, the others are not. 1.2.3 (5) I do not like fruit, or I like cereal. (6) I do not like fruit, or I do not like cereal. (7) I know how to cook an omelette and I like fruit, or I do not like cereal. (8) I know how to cook an omelette, and I like fruit or I do not like cereal. 1.2.6 (1) F G. (2) E (G F). (4) G F. 1.2.8 X = F, Y = T, Z = F, W=T (1) Z Y is True F T T (2) X Z is True F T F (3) (Y W) X is False T T T F F (4) W (X W) is True T F T F (5) [(Y W) W] X is True T T T T T T T (6) (W X) (Z Y) is True T F F T F F T T
(1) False: Flora doesn’t like carrots (2) True: (T or F), and T is true (3) True: F or (T and T) is true (4) False: (T or T) and (F or F) is false (5) False: F or (T and (F or F)) is false 1.3.6 In each of these, use truth tables to check that the statement corresponding to the meta statement is a tautology. (ii) P Q (P Q) Q P T T F F
T
F
T
F
T F F T F
F F T T F
T F F T T
T T T T T
(iii) P Q (P Q) P T T F F
T
F
T
F
T T T
F T T
F T F
F T F
(v) P Q P (P Q) T T F F
T
F
T
F
T T T
T T T
F T T
F T F
(vii) P Q (P Q) P Q T T F F
T
F
T
F
T F F T T
T F F T F
T T T T T
F F T T F
(xii) P Q (P Q) P Q T T F F
T
F
T
F
F F
T T
T T
T T
(xiv) P Q (P Q) P Q T T F F
T
F
T
F
F F
T T
F F
F F
1.3.8 (2) The original statement is: “I will go home if it is after midnight.” Write this as A B where A = “It is after midnight” and B=”I will go home.” Converse (B A): It is after midnight if I will go home. Inverse (A B): I will not go home if it is not after midnight. Contrapositive (B A): It is not after midnight if I will not go home. (3) Converse: Good neighbors make good fences. Inverse: Bad fences make bad neighbors. Contrapositive: Bad neighbors make bad fences. 1.3.9 (1) Inverse; (2) None; (3) Converse/None?; (4) Contrapositive; (5) None; (6) Inverse. 1.3.10 (1) e^5 0. (2) 3 5 and 7 < 8. [Use Fact 1.3.2(xiii)] (3) sin(/2) 0 or tan(0) < 0. [Use Fact 1.3.2(xii)] (4) y = 3 and y^2 7. [Use Fact 1.3.2(xiv)] (5) w – 3 > 0 and w^2 + 9 6w. [Use Fact 1.3.2(xiv)] (6) a – b = c and a b+c, or a-b c and a = b+c. [Use Fact 1.3.2(xv)] 1.3.11 (1) It is not Monday, or it is not snowing. [Use Fact 1.3.2(xii)] (3) Susan doesn’t like to eat figs, or she doesn’t like to drink prune juice. [Use Fact 1.3.2(xiii)] (5) The play will end on time and the actors are not in good spirits, or the play will not end on time and the actors are in good spirits. [Use Fact 1.3.2(xv)]
