20
Formal Logic
c. Unhappy bears means that the fish are not plentiful and also that there is heavy snow.
d. Unhappy bears are a necessary condition for heavy snow.
e. The snow is heavy if and only if the fish are not plentiful.
22. Using letters P, C, B, L for the component statements, translate the following compound statements into
symbolic notation.
a. If the project is finished soon, then the client will be happy and the bills will be paid.
b. If the bills are not paid, then the lights will go out.
c. The project will be finished soon only if the lights do not go out.
d. If the bills are not paid and the lights go out, then the client will not be happy.
e. The bills will be paid if and only if the project is finished soon, or else the lights go out.
f. The bills will be paid if and only if either the project is finished soon or the lights go out.
23. Construct truth tables for the following wffs. Note any tautologies or contradictions.
a. (A S B) 4 A′ ~ B
d. A ` B S A′
b. (A ` B) ~ C S A ` (B ~ C )
e. (A S B) S [(A ~ C ) S (B ~ C )]
c. A ` (A′ ~ B′)′
24. Construct truth tables for the following wffs. Note any tautologies or contradictions.
a. A S (B S A)
d. [(A ~ B) ` C′] S A′ ~ C
b. A ` B 4 B′ ~ A′
e. A′ S (B ~ C′)
c. (A ~ B′) ` (A ` B)′
25. Verify the equivalences in the list on page 9 by constructing truth tables. (We have already verified
1a, 4b, and 5a.)
26. Verify by constructing truth tables that the following wffs are tautologies. Note that the tautologies in parts
b, e, f, and g produce equivalences such as (A′)′ 3 A.
a. A ~ A′
e. (A ~ B)′ 4 A′ ` B′ (De Morgan’s law)
b. (A′)′ 4 A
f. (A ` B)′ 4 A′ ~ B′ (De Morgan’s law)
c. A ` B S B
g. A ~ A 4 A
d. A S A ~ B
27. Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that
will convert the left side into the right side. You may use any of the equivalencies in the list on page 9 or
the equivalencies from Exercise 26.
a. (A ` B′) ` C 4 (A ` C ) ` B′
b. (A ~ B) ` (A ~ B′) 4 A
c. A ~ (B ` A′) 4 A ~ B
28. Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that
will convert the left side into the right side. You may use any of the equivalencies in the list on page 9 or
the equivalencies from Exercise 26.
a. (A ` B′)′ ~ B 4 A′ ~ B
b. A ` (A ` B′)′ 4 A ` B
c. (A ` B)′ ` (A ~ B′) 4 B′
29. We mentioned that (A ` B) ~ C cannot be proved equivalent to A ` (B ~ C ) using either of the associative
tautological equivalences, but perhaps it can be proved some other way. Are these two wffs equivalent?
Prove or disprove.
Formal Logic
c. Unhappy bears means that the fish are not plentiful and also that there is heavy snow.
d. Unhappy bears are a necessary condition for heavy snow.
e. The snow is heavy if and only if the fish are not plentiful.
22. Using letters P, C, B, L for the component statements, translate the following compound statements into
symbolic notation.
a. If the project is finished soon, then the client will be happy and the bills will be paid.
b. If the bills are not paid, then the lights will go out.
c. The project will be finished soon only if the lights do not go out.
d. If the bills are not paid and the lights go out, then the client will not be happy.
e. The bills will be paid if and only if the project is finished soon, or else the lights go out.
f. The bills will be paid if and only if either the project is finished soon or the lights go out.
23. Construct truth tables for the following wffs. Note any tautologies or contradictions.
a. (A S B) 4 A′ ~ B
d. A ` B S A′
b. (A ` B) ~ C S A ` (B ~ C )
e. (A S B) S [(A ~ C ) S (B ~ C )]
c. A ` (A′ ~ B′)′
24. Construct truth tables for the following wffs. Note any tautologies or contradictions.
a. A S (B S A)
d. [(A ~ B) ` C′] S A′ ~ C
b. A ` B 4 B′ ~ A′
e. A′ S (B ~ C′)
c. (A ~ B′) ` (A ` B)′
25. Verify the equivalences in the list on page 9 by constructing truth tables. (We have already verified
1a, 4b, and 5a.)
26. Verify by constructing truth tables that the following wffs are tautologies. Note that the tautologies in parts
b, e, f, and g produce equivalences such as (A′)′ 3 A.
a. A ~ A′
e. (A ~ B)′ 4 A′ ` B′ (De Morgan’s law)
b. (A′)′ 4 A
f. (A ` B)′ 4 A′ ~ B′ (De Morgan’s law)
c. A ` B S B
g. A ~ A 4 A
d. A S A ~ B
27. Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that
will convert the left side into the right side. You may use any of the equivalencies in the list on page 9 or
the equivalencies from Exercise 26.
a. (A ` B′) ` C 4 (A ` C ) ` B′
b. (A ~ B) ` (A ~ B′) 4 A
c. A ~ (B ` A′) 4 A ~ B
28. Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that
will convert the left side into the right side. You may use any of the equivalencies in the list on page 9 or
the equivalencies from Exercise 26.
a. (A ` B′)′ ~ B 4 A′ ~ B
b. A ` (A ` B′)′ 4 A ` B
c. (A ` B)′ ` (A ~ B′) 4 B′
29. We mentioned that (A ` B) ~ C cannot be proved equivalent to A ` (B ~ C ) using either of the associative
tautological equivalences, but perhaps it can be proved some other way. Are these two wffs equivalent?
Prove or disprove.