Symbolic Logic 5E: 2.4, II

“Use truth tables to decide which of the following are logical equivalences”

1. Not equivalent.

P

Q

¬P

¬Q

P→Q

¬P→¬Q

T

T

F

F

T

T

T

T

F

F

T

F

T

F

F

F

T

T

T

T

T

F

T

T

F

T

F

F

2. Equivalent.

P

Q

¬P

¬Q

P→Q

¬Q→¬P

T

T

F

F

T

T

T

T

F

F

T

F

F

T

F

F

T

T

T

T

T

F

T

T

F

T

T

T

3. Equivalent

P

Q

R

P ∧ Q

(P ∧ Q)→R

P→ (Q→R)

Q→R

T

T

T

T

T

T

T

T

T

T

F

T

F

F

F

T

T

F

F

F

F

T

T

T

T

F

T

F

T

T

T

T

F

F

F

F

T

T

T

T

F

F

T

F

T

T

T

T

F

T

T

F

T

T

T

T

F

T

F

F

T

T

F

T

4. Not equivalent.

P

Q

R

P→Q

Q→R

P→ (Q→R)

(P→Q)→R

T

T

T

T

T

T

T

T

T

T

F

T

F

F

F

T

T

F

F

F

T

T

T

T

T

F

T

F

T

T

T

T

F

F

F

T

T

T

F

F

F

F

T

T

T

T

T

T

F

T

T

T

T

T

T

T

F

T

F

T

F

T

F

F

5. Equivalent.

P

Q

R

Q ∨ R

P ∧ Q

P ∧ R

P ∧ (Q ∨ R)

(P ∧ Q) ∨ (P ∧ R)

T

T

T

T

T

T

T

T

T

T

T

F

T

T

F

T

T

T

T

F

F

F

F

F

F

F

T

T

F

T

T

F

T

T

T

T

F

F

F

F

F

F

F

F

T

F

F

T

T

F

F

F

F

T

F

T

T

T

F

F

F

F

T

F

T

F

T

F

F

F

F

T

6. Not equivalent.

P

Q

R

Q ∧ R

P ∨ Q

P ∨ ( Q ∧ R)

(P ∨ Q) ∧ R

T

T

T

T

T

T

T

T

T

T

F

F

T

T

F

F

T

F

F

F

T

T

F

F

T

F

T

F

T

T

T

T

F

F

F

F

F

F

F

T

F

F

T

F

F

F

F

T

F

T

T

T

T

T

T

T

F

T

F

F

T

F

F

T

7. Equivalent.

P

Q

R

Q ∧ R

P ∨ ( Q ∧ R)

P ∨ Q

P ∨ R

(P ∨ Q) ∧ (P ∨ R)

T

T

T

T

T

T

T

T

T

T

T

F

F

T

T

T

T

T

T

F

F

F

T

T

T

T

T

T

F

T

F

F

T

T

T

T

F

F

F

F

F

F

F

F

T

F

F

T

F

F

F

T

F

T

F

T

T

T

T

T

T

T

T

F

T

F

F

F

T

F

F

T

8. Equivalent.

P

Q

P≡Q

P ∧ Q

¬P

¬Q

¬P ∧ ¬Q

(P ∧ Q) ∨ (¬P ∧ ¬Q)

T

T

T

T

F

F

F

T

T

T

F

F

F

F

T

F

F

T

F

F

T

F

T

T

T

T

T

F

T

F

F

T

F

F

F

T

9. Not equivalent.

P

Q

P→Q

P ∧ (P→Q)

T

T

T

T

T

T

F

F

F

F

F

F

T

F

T

F

T

T

F

T

10. Equivalent.

P

Q

Q→P

P ∧ (Q→P)

T

T

T

T

T

T

F

T

T

T

F

F

T

F

T

F

T

F

F

T

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