"The converse, inverse, and contrapositive of a conditional statement"

Conditional Statement Structure

• A conditional statement has the form:
  • "If P, then Q."
  • P: Antecedent (condition)
  • Q: Consequent (result)

Converse, Inverse, and Contrapositive Definitions

• Converse: Reverse the roles of the antecedent and consequent.

  • Form: "If Q, then P."

• Inverse: Negate both the antecedent and consequent.

  • Form: "If not P, then not Q."

• Contrapositive: Switch and negate both the antecedent and consequent.

  • Form: "If not Q, then not P."

Example Conditional Statement

• Given Statement:
  • If a class is offered at Princeton University, then the class is offered at a U.S. university.

Identification of Antecedent and Consequent

• Antecedent: A class is offered at Princeton University
• Consequent: The class is offered at a U.S. university

Finding Converse, Inverse, and Contrapositive

1. Converse:

   • If a class is offered at a U.S. university, then the class is offered at Princeton University.
   • Truth Value: False

2. Inverse:

   • If a class is not offered at Princeton University, then the class is not offered at a U.S. university.
   • Truth Value: False

3. Contrapositive:

   • If a class is not offered at a U.S. university, then the class is not offered at Princeton University.
   • Truth Value: True

Summary of Results

• Original Statement: True
• Converse: False
• Inverse: False
• Contrapositive: True

Symbolic Representation

• Let p = "A class is offered at Princeton University"
• Let q = "The class is offered at a U.S. university"
  • The original statement: p → q
  • Converse: q → p
  • Inverse: ¬p → ¬q
  • Contrapositive: ¬q → ¬p