Construct a truth table for the statement.
(p ↔ q) → p
To construct a truth table for the given statement (p ↔ q) → p, we need to consider all possible combinations of truth values for p and q, and determine the resulting truth value for the entire statement.
Let's start by examining the components of the statement:
1. (p ↔ q): This represents the logical biconditional, also known as "if and only if." It is true if p and q have the same truth values, and false otherwise. We can represent this relationship using the following truth table:
| p | q | p ↔ q |
| --- | --- | ------- |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
2. → : This represents the conditional "if...then." It is false when the antecedent (p ↔ q) is true and the consequent (p) is false, and true in all other cases. We can represent this using the following truth table:
| p ↔ q | p | (p ↔ q) → p |
| ------- | --- | ------------- |
| T | T | T |
| F | T | T |
| F | F | T |
| T | F | F |
Combining the two truth tables and evaluating the overall statement (p ↔ q) → p, we can construct the complete truth table:
| p | q | p ↔ q | (p ↔ q) → p |
| --- | --- | ------- | ------------- |
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | T | F |
This is the truth table for the given statement (p ↔ q) → p. It displays all possible combinations of truth values for p and q, and the resulting truth value for the entire statement.