“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 |