Construct a truth table for the statement.
(p ↔ q) → p
p. q. p<->q
T. T. T
T. F. F
F. T. F
F. F. T
thank you!
To construct a truth table for the statement "(p ↔ q) → p", we need to consider all possible combinations of truth values for the propositions p and q.
First, let's list all possible combinations of truth values for p and q. Since each proposition can take on either a true (T) or false (F) value, there are four possible combinations:
1. p = T, q = T
2. p = T, q = F
3. p = F, q = T
4. p = F, q = F
Now, we can substitute these truth values into the statement "(p ↔ q) → p" and determine the resulting truth value for each combination.
For combination 1 (p = T, q = T):
- (p ↔ q) → p
- (T ↔ T) → T
- T → T
- T
For combination 2 (p = T, q = F):
- (p ↔ q) → p
- (T ↔ F) → T
- F → T
- T
For combination 3 (p = F, q = T):
- (p ↔ q) → p
- (F ↔ T) → F
- F → F
- T
For combination 4 (p = F, q = F):
- (p ↔ q) → p
- (F ↔ F) → F
- T → F
- F
Therefore, the truth table for the statement "(p ↔ q) → p" is as follows:
| p | q | (p ↔ q) → p |
|---|---|-------------|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
In this truth table, the column "(p ↔ q) → p" represents the resulting truth value of the statement for each combination of p and q.