2.4 Biconditional Statements

Logic

• The statement $P\to Q$ doesn’t necessarily mean that $Q\to P$. Suppose that $a$ is some integer and consider the statements:
  • (a is a multiple of 6) $\to$ (a is divisible by 2),
  • (a is divisible by 2) $\to$ (a is a multiple of 6).
    • The first statement is true while the second one is false.
• Sometimes the converse is true.
  • (a is even) $\to$ (a is divisible by 2),
  • (a is divisible by 2) $\to$ (a is even)
    • Since both $P\to Q$ and$Q\to P$ are true, it follows that ($P\to Q$) $\wedge$ ($Q\to P$) is true.
• The symbol for a biconditional statement is $⟺$ which is read as if and only if
  • E.g. $P⟺Q$

Truth Table

$P$	$Q$	$P⟺Q$
T	T	T
T	F	F
F	T	F
F	F	T