Biconditionals

The converse

Given an implication \(P\implies Q\), its converse is the statement \(Q\implies P\).

Statement and Converse are different

If I own a BMW 335xi, then I own a car

  • \(P\) is “I own a BMW 335xi”
  • \(Q\) is “I own a car”

The converse is “If I own a car, then I own a BMW 335xi”.

\(P\implies Q\) is true but \(Q\implies P\) is false.

Biconditionals or Equivalence

\(P\iff Q\) means “If P, then Q” AND “If Q, then P”. It is often read “if and only if” since

  • \(P\) if \(Q\) means \(Q\implies P\)
  • \(P\) only if \(Q\) means \(P\implies Q\).

It can also be read “necessary and sufficient” (\(P\) is necessary and sufficient for \(Q\)).

Truth Table for Equivalence

Synonyms

  • \(P\) if and only if \(Q\)
  • \(P\) is necessary and sufficient for \(Q\)
  • \(P\) is equivalent to \(Q\)
  • If \(P\), then \(Q\), and conversely.

Sample problem

Put the statement “If \(xy=0\) then \(x=0\) or \(y=0\), and conversely” in the form “P if and only if Q”.