Section 1.1 Statements, Symbolic Representation, and Tautologies
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c. Whenever violets are blue, roses are red and sugar is sweet.
d. Roses are red only if violets aren’t blue or sugar is sour.
e. Roses are red and, if sugar is sour, then either violets aren’t blue or sugar is sweet.
16. Let A, B, and C, and D be the following statements:
A The villain is French.
B The hero is American.
C The heroine is British.
D The movie is good.
Translate the following compound statements into symbolic notation.
a. The hero is American and the movie is good.
b. Athough the villain is French, the movie is good.
c. If the movie is good, then either the hero is American or the heroine is British.
d. The hero is not American, but the villain is French.
e. A British heroine is a necessary condition for the movie to be good.
17. Use A, B, and C as defined in Exercise 15 to translate the following statements into English.
a. B ~ C′
e. (B ` C′)′ S A
b. B′ ~ (A S C )
f. A ~ (B ` C′)
c. (C ` A′) 4 B
g. (A ~ B) ` C′
d. C ` (A′ 4 B)
18. Use A, B, and C as defined in Exercise 16 to translate the following statements into English.
a. B S A′
e. A 4 (B ~ C )
b. B ` C ` D′
f. D′ S (A ~ C )′
c. B S (C ~ A)
g. (C S D) ` (A S B′)
d. (A ~ C ) S B′
19. Using letters H, K, A for the component statements, translate the following compound statements into
symbolic notation.
a. If the horse is fresh, then the knight will win.
b. The knight will win only if the horse is fresh and the armor is strong.
c. A fresh horse is a necessary condition for the knight to win.
d. The knight will win if and only if the armor is strong.
e. A sufficient condition for the knight to win is that the armor is strong or the horse is fresh.
20. Using letters A, T, E for the component statements, translate the following compound statements into
symbolic notation.
a. If Anita wins the election, then tax rates will be reduced.
b. Tax rates will be reduced only if Anita wins the election and the economy remains strong.
c. Tax rates will be reduced if the economy remains strong.
d. A strong economy will follow from Anita winning the election.
e. The economy will remain strong if and only if Anita wins the election or tax rates are reduced.
21. Using letters F, B, S for the component statements, translate the following compound statements into
symbolic notation.
a. Plentiful fish are a sufficient condition for bears to be happy.
b. Bears are happy only if there are plentiful fish.
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c. Whenever violets are blue, roses are red and sugar is sweet.
d. Roses are red only if violets aren’t blue or sugar is sour.
e. Roses are red and, if sugar is sour, then either violets aren’t blue or sugar is sweet.
16. Let A, B, and C, and D be the following statements:
A The villain is French.
B The hero is American.
C The heroine is British.
D The movie is good.
Translate the following compound statements into symbolic notation.
a. The hero is American and the movie is good.
b. Athough the villain is French, the movie is good.
c. If the movie is good, then either the hero is American or the heroine is British.
d. The hero is not American, but the villain is French.
e. A British heroine is a necessary condition for the movie to be good.
17. Use A, B, and C as defined in Exercise 15 to translate the following statements into English.
a. B ~ C′
e. (B ` C′)′ S A
b. B′ ~ (A S C )
f. A ~ (B ` C′)
c. (C ` A′) 4 B
g. (A ~ B) ` C′
d. C ` (A′ 4 B)
18. Use A, B, and C as defined in Exercise 16 to translate the following statements into English.
a. B S A′
e. A 4 (B ~ C )
b. B ` C ` D′
f. D′ S (A ~ C )′
c. B S (C ~ A)
g. (C S D) ` (A S B′)
d. (A ~ C ) S B′
19. Using letters H, K, A for the component statements, translate the following compound statements into
symbolic notation.
a. If the horse is fresh, then the knight will win.
b. The knight will win only if the horse is fresh and the armor is strong.
c. A fresh horse is a necessary condition for the knight to win.
d. The knight will win if and only if the armor is strong.
e. A sufficient condition for the knight to win is that the armor is strong or the horse is fresh.
20. Using letters A, T, E for the component statements, translate the following compound statements into
symbolic notation.
a. If Anita wins the election, then tax rates will be reduced.
b. Tax rates will be reduced only if Anita wins the election and the economy remains strong.
c. Tax rates will be reduced if the economy remains strong.
d. A strong economy will follow from Anita winning the election.
e. The economy will remain strong if and only if Anita wins the election or tax rates are reduced.
21. Using letters F, B, S for the component statements, translate the following compound statements into
symbolic notation.
a. Plentiful fish are a sufficient condition for bears to be happy.
b. Bears are happy only if there are plentiful fish.