Write the converse, inverse, and contrapositive of the conditional statement. Find the truth value of each. If John is on vacation, then he is in London. Converse: If John is not on vacation, then he is not in London. Inverse: If John is in London, then he is on vacation. Contrapositive: If John is not in London, he is not on vacation. Converse: If John is in London, then he is on vacation. Inverse: If John is on vacation, then he is not in London. Contrapositive: If John is not in London, he is on vacation. Converse: If John is in London, then he is on vacation. Inverse: If John is not in London, he is not on vacation. Contrapositive: If John is not on vacation, then he is not in London. Converse: If John is in London, then he is on vacation. Inverse: If John is not on vacation, then he is not in London. Contrapositive: If John is not in London, he is not on vacation. If

we analyze the original statement and its variations, we can determine the truth values as follows: Original Statement: If John is on vacation, then he is in London. Let's assign variables to the statements: P: John is on vacation. Q: John is in London. The truth table for the original statement is as follows: | P | Q | P → Q | |---|---|-------| | T | T | T | | T | F | F | | F | T | T | | F | F | T | Now let's analyze the truth values of the converse, inverse, and contrapositive statements: Converse: If John is not on vacation, then he is not in London. Let's assign variables to the converse statement: P': John is not on vacation. Q': John is not in London. The truth table for the converse statement is as follows: | P' | Q' | P' → Q' | |---|---|-------| | T | T | T | | T | F | F | | F | T | T | | F | F | T | Inverse: If John is in London, then he is on vacation. Let's assign variables to the inverse statement: P'': John is in London. Q'': John is on vacation. The truth table for the inverse statement is as follows: | P'' | Q'' | P'' → Q'' | |---|---|-------| | T | T | T | | T | F | F | | F | T | T | | F | F | T | Contrapositive: If John is not in London, then he is not on vacation. Let's assign variables to the contrapositive statement: P''': John is not in London. Q''': John is not on vacation. The truth table for the contrapositive statement is as follows: | P''' | Q''' | P''' → Q''' | |---|---|-------| | T | T | T | | T | F | F | | F | T | T | | F | F | T | Now, let's determine the truth values of each statement: Original Statement: If John is on vacation, then he is in London. The truth value is T (true) because if John is on vacation, it is true that he is in London. Converse: If John is not on vacation, then he is not in London. The truth value is F (false) because it is possible for John to be in London without being on vacation (e.g., visiting friends). Inverse: If John is in London, then he is on vacation. The truth value is F (false) because it is possible for John to be in London without being on vacation (e.g., visiting friends). Contrapositive: If John is not in London, then he is not on vacation. The truth value is T (true) because if John is not in London, it is true that he is not on vacation.