8th grade Q2

The equivalence of a statement and its contrapositive means they are logically the same—both are true or false under the same conditions.

To illustrate this, you can use a truth table to compare the truth values of the original statement ($p\to q$) and its contrapositive ($\neg q\to \neg p$).

Truth Table for Equivalence

The truth table evaluates all possible combinations of the hypothesis ($p$) and conclusion ($q$). For the an if-then statement and it's contrapositive, the truth table is:

Explanation of the Table

1. Row 1: When both $p$ and $q$ are true, $p\to q$ and $\neg q\to \neg p$ are true.
2. Row 2: When $p$ is true and $q$ is false, both $p\to q$ and $\neg q\to \neg p$ are false.
3. Row 3: When $p$ is false and $q$ is true, both $p\to q$ and $\neg q\to \neg p$ are true.
4. Row 4: When both $p$ and $q$ are false, both $p\to q$ and $\neg q\to \neg p$ are true.

Since $p\to q$ and $\neg q\to \neg p$ have identical truth values in all rows, they are logically equivalent.

Example

Original statement: "If it rains, then the ground will be wet."

• Hypothesis: "It rains" ($p$).
• Conclusion: "The ground will be wet" ($q$).

Contrapositive: "If the ground is not wet, then it did not rain."

• Hypothesis: "The ground is not wet" ($\neg q$).
• Conclusion: "It did not rain" ($\neg p$).

Note

Both the original statement and its contrapositive are true. This is because statements and their contrapositives are logically equivalent.

Key Points:

• A statement ($p\to q$) and its contrapositive ($\neg q\to \neg p$) are logically equivalent.
• Truth tables provide a systematic way to verify this equivalence.
• When doing a mathematical proof, sometimes it's more convenient to prove the contrapositive of a statement.