Statements of the form, "IF P THEN Q" are called implications. P is frequently called the hypotheses or antecedent, and Q is called the conclusion or consequent. Other related expressions are:
IF Q THEN P. This is called the converse.
IF {not Q} THEN {not P}. This is called the contrapositive.
IF (not P) THEN (not Q). This is called the inverse.
Example

• Original implication: If pigs are green, then I can fly. So "P" is "pigs are green" and "Q" is "I can fly."
• Converse: If I can fly, then pigs are green.
• Contrapositive: If I can't fly, then pigs aren't green.
• Inverse: If pigs aren't green, then I can't fly.

• Implication and converse both true
  Implication: If x³ is a positive real number, then x is a positive real number. True!
  Converse: If x is a positive real number, then x³ is a positive real number. True!
  Contrapositive: If x is not a positive real number, then x³ is not a positive real number. This is True!, and has the same logical "content" as the original implication (although to me it is more difficult to understand!).
• Implication true and converse false
  Implication: If x is an integer, then x is a real number. True!
  Converse: If x is a real number, then x is an integer. False!
  Contrapositive: If x is not a real number, then x is not an integer. Also True!

Maintained by greenfie@math.rutgers.edu and last modified 3/27/2009.