In a truth table, the compound statement 'if-then', also known as a conditional statement, is expressed using the logical implication operator. It is typically represented as \( P \rightarrow Q \), where \( P \) is the antecedent (the 'if' part) and \( Q \) is the consequent (the 'then' part).
The truth table for the implication \( P \rightarrow Q \) is as follows:
| \( P \) | \( Q \) | \( P \rightarrow Q \) | |---------|---------|-----------------------| | T | T | T | | T | F | F | | F | T | T | | F | F | T |
In this table:
- The implication is only false when \( P \) is true and \( Q \) is false.
- In all other cases (when \( P \) is false or both \( P \) and \( Q \) are true), the implication is true.
Thus, the 'if-then' statement is an expression of logical implication in a truth table.