Eduprobe
Question:
If (p ∧ ¬q) ∧ (p ∧ r) → ¬p ∨ q is false, then the truth values of p, q, and r are, respectively.
F,T,F
T,T,T
F,F,F
T,F,T
Solution:
The truth table for the (p ∧ ¬q) ∧ (p ∧ r) → ¬p ∨ q is:
| p | q | r | ¬q | p ∧ ¬q | p ∧ r | (p ∧ ¬q) ∧ (p ∧ r) | ¬p | ¬p ∨ q | (p ∧ ¬q) ∧ (p ∧ r) → ¬p ∨ q |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
Only when p is true, q is false, and r is true, the expression (p ∧ ¬q) ∧ (p ∧ r) → ¬p ∨ q is false. Therefore, the truth values of p, q, and r are, respectively, T, F, T.