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Consider the conditional statement "if P, then Q." Its contrapositive is "if not Q, then not P." We will work with these two statements and their truth values.

A truth table for P and Q should contain all possible combinations of truth values for P and Q. Here's the table: | P | Q | |---|---| | T | T | | T | F | | F | T | | F | F |

Remember that the conditional statement is "if P, then Q" and its contrapositive is "if not Q, then not P." We'll first determine the truth values for the conditional statement. The conditional statement is false only when P is true and Q is false; otherwise, it is true. | P | Q | If P, then Q | |---|---|--------------| | T | T | T | | T | F | F | | F | T | T | | F | F | T | Next, we'll determine the truth values for the contrapositive, "if not Q, then not P." Since the contrapositive is just the negation of both P and Q in the conditional statement, we can represent not P as ¬P and not Q as ¬Q. The contrapositive is false only when ¬Q is true and ¬P is false; otherwise, it is true. | ¬Q | ¬P | If not Q, then not P | |----|----|---------------------| | F | F | T | | F | T | T | | T | F | F | | T | T | T |

Now, let's combine the two tables and compare the truth values of the conditional statement and its contrapositive: | P | Q | If P, then Q | ¬Q | ¬P | If not Q, then not P | |---|---|--------------|----|----|---------------------| | T | T | T | F | F | T | | T | F | F | F | T | T | | F | T | T | T | F | F | | F | F | T | T | T | T | As we can see, the truth values of the conditional statement ("if P, then Q") and its contrapositive ("if not Q, then not P") are the same for all combinations of truth values of P and Q. Therefore, a conditional statement and its contrapositive are logically equivalent to each other.