SOLUTION: Find the truth value of the following: Show the solution. 3. [(q∨¬p)↔(r→p)]→(q∨s)

Question 1181634: Find the truth value of the following: Show the solution.

3. [(q∨¬p)↔(r→p)]→(q∨s)

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's determine the truth value of the statement [(q∨¬p)↔(r→p)]→(q∨s) using a truth table. Since there are four variables (p, q, r, s), there are 2⁴ = 16 possible combinations of truth values.
Here's the truth table:
| p | q | r | s | ¬p | q∨¬p | r→p | (q∨¬p)↔(r→p) | q∨s | [(q∨¬p)↔(r→p)]→(q∨s) |
|---|---|---|---|----|------|-----|----------------|-----|--------------------------|
| T | T | T | T | F | T | T | T | T | T |
| T | T | T | F | F | T | T | T | T | T |
| T | T | F | T | F | T | T | T | T | T |
| T | T | F | F | F | T | T | T | T | T |
| T | F | T | T | F | F | T | F | T | T |
| T | F | T | F | F | F | T | F | F | T |
| T | F | F | T | F | F | T | F | T | T |
| T | F | F | F | F | F | T | F | F | T |
| F | T | T | T | T | T | F | F | T | T |
| F | T | T | F | T | T | F | F | T | T |
| F | T | F | T | T | T | T | T | T | T |
| F | T | F | F | T | T | T | T | T | T |
| F | F | T | T | T | T | F | F | T | T |
| F | F | T | F | T | T | F | F | F | T |
| F | F | F | T | T | T | T | T | T | T |
| F | F | F | F | T | T | T | T | F | F |
Since the final column has both T and F values, the given statement is a **contingency**. Its truth value depends on the specific truth values of p, q, r, and s.

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