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 Ú F).
(4) Ø 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) Ù X is False
T T T F F
(4) W ® (X ® Ø W) is True
T F T F
(5) [(Y ® W) « W] ÙØ X is True
T T T T T T T
(6) (W ® X) « Ø (Z Ú Y) is True
T F F T F F T T
1.2.9
(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) Ù Ø Q ® ØPT
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 Ù Q) ® PT
T
F
F / T
F
T
F / T T T
F T T
F T F
F T F
(v)
P / Q / P ® (P Ú 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 Ú Q) Ù Ø P ® QT
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
(xi)
T
T
F
F / T
F
T
F / T T T T T
F F T T F
T F F T F
T T T T T
1.3.7 These are abbreviated tables, with only final value included on each side. Observe that the truth values on both sides agree to verify the tautology.
(iv)
P / Q / R / (P Ú Q) Ú R « P Ú (Q Ú R)T
T
T
T
F
F
F
F / T
T
F
F
T
T
F
F / T
F
T
F
T
F
T
F / T T
T T
T T
T T
T T
T T
T T
F F
(vii)
P / Q / R / P Ú (Q Ù R ) « (P Ú Q) Ù (P Ú R)T
T
T
T
F
F
F
F / T
T
F
F
T
T
F
F / T
F
T
F
T
F
T
F / T T
T T
T T
T T
T T
F F
F F
F F
(ix)
P / Q / P ® Q « Ø Q ® ØPT
T
F
F / T
F
T
F / T T
F F
T T
T T
(xii)
T
T
F
F / T
F
T
F / F F
T T
T T
T T
(xiv)
P / Q / Ø (P ® Q) « P Ù ØQT
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 (ØA ® ØB): I will not go home if it is not after midnight.
Contrapositive (ØB ® ØA): 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) e5 £ 0.
(2) 3 ³ 5 and 7 < 8. [Use Fact 1.3.2(xiii)]
(3) sin(p/2) ³ 0 or tan(0) < 0. [Use Fact 1.3.2(xii)]
(4) y = 3 and y2¹ 7. [Use Fact 1.3.2(xiv)]
(5) w – 3 > 0 and w2+ 9 £ 6w. [Use Fact 1.3.2(xiv)]
(6) a – b = c and a ¹ b+c, or a-b ¹ 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)]