8th grade Q2

The converse and inverse of a statement are logically equivalent, meaning they always have the same truth values under all conditions. To illustrate this equivalence, you can use a truth table to compare their truth values.

Definitions

1. Original Statement: "If $p$, then $q$" ($p\to q$).
2. Converse: "If $q$, then $p$" ($q\to p$).
3. Inverse: "If not $p$, then not $q$" ($\neg p\to \neg q$).

Truth Table

The following truth table evaluates all possible combinations of $p$ and $q$ and compares the truth values of the converse ($q\to p$) and inverse ($\neg p\to \neg q$).

Explanation of the Table:

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

Since $q\to p$ and $\neg p\to \neg q$ 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" ($q$).

Converse: "If the ground is wet, then it rained."

• Hypothesis: "The ground is wet" ($q$).
• Conclusion: "It rained" ($p$).

Inverse: "If it does not rain, then the ground will not be wet."

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

Note

• The converse is not always true: Just because the ground is wet doesn't mean it is wet due to the rain.
• The inverse is not always true: Just because it did not rain you can't conclude the ground is not wet.

Key Points:

• The converse ($q\to p$) and inverse ($\neg p\to \neg q$) of a statement are always logically equivalent.