8th grade Q2

The contrapositive of an if-then statement is formed by switching the hypothesis and conclusion and negating both.

If the original statement is

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

the contrapositive is

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

Steps to Determine the Contrapositive

1. Start with the Original Statement:
   Write the statement in the form "If $p$, then $q$."
2. Switch the Hypothesis and Conclusion:
   Rewrite the statement as "If $q$, then $p$."
3. Negate Both Parts:
   Negate the new hypothesis and conclusion to form "If not $q$, then not $p$."

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" ($\neg p$).

Key Points

• The contrapositive switches and negates both the hypothesis and the conclusion.