Notes on Conditional Statements

Introduction to Conditional Statements

• Conditional Statement: Presented as "If P, then Q."
  • Example: "If you live in Los Angeles, then you live in California."
  • Hypothesis (P): The part after 'if'
  • Conclusion (Q): The part after 'then'
• Negation Symbol: Represents 'not P'.
  • Example: "It is not sunny outside."

Converse, Inverse, and Contrapositive

Converse

• Definition: Reverse of the conditional statement.
  • Structure: "If Q, then P."
  • Example: "If you live in California, then you live in Los Angeles."
  • Truth Value: Converse is not always true.
  • Biconditional Statement: Occurs if both conditional and converse are true.

Inverse

• Definition: Negation of the conditional statement.
  • Structure: "If not P, then not Q."
  • Example: "If you don't live in Los Angeles, then you don't live in California."
  • Truth Value: Inverse shares the same truth value as the converse.

Contrapositive

• Definition: Reverse negation of the conditional statement.
  • Structure: "If not Q, then not P."
  • Example: "If you don't live in California, then you don't live in Los Angeles."
  • Truth Value: Contrapositive shares the same truth value as the conditional statement.

Example Problem

• Conditional Statement: "If I am hungry, then I will eat pizza."
  • Converse: "If I eat pizza, then I am hungry."
  • Inverse: "If I am not hungry, then I will not eat pizza."
  • Contrapositive: "If I don't eat pizza, then I am not hungry."

Key Points

• Conditional statements and their contrapositives have the same truth value.
• Converse and inverse share the same truth value.
• Understanding the structure and truth value of each form is crucial for logical reasoning.