8th grade Q2

The inverse of an if-then statement reverses the truth value of both the hypothesis and the conclusion without changing their order.

If the original statement is

"If $p$, then $q$" ($p\to q$),

the inverse is

"If not $p$, then not $q$" ($\neg p\to \neg q$).

Steps to Determine the Inverse

1. Start with the Original Statement:
   Write the statement in the form "If $p$, then $q$."
2. Negate Both Parts:
   Negate the hypothesis: Replace $p$ with not $p$ ($\neg p$).
   Negate the conclusion: Replace $q$ with not $q$ ($\neg q$).
3. Rewrite as If-Then:
   Combine the negated parts into the form "If not $p$, then not $q$."

Example

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

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

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

• Hypothesis: "It does not rain" ($\neg p$).
• Conclusion: "The ground will not be wet" ($\neg q$).

Key Points

• The inverse is formed by negating both the hypothesis and conclusion of the original statement.
• The inverse is not logically equivalent to the original statement. That is to say, if the original statement is true, the inverse cannot be inferred to be true.