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Biconditional or Equivalence

A statement of the form “p if and only if q” is called “biconditional” (pq being logical statements) or “double implication”. Symbolically “p iff q” is represented by p  q or by p  q. Biconditional also means
  1. p is a necessary and sufficient condition for q.
  2. q is a necessary and sufficient condition for p.
  3. If p then q and if q then p.
  4. q if and only if p.
For example, if p: he will be successful and q: he works hard, then p  q means. “He will be successful if and only if he works hard” or “Hard work is a necessary and sufficient condition for success”. The following are other illustrations which actually do not appear to be so but they infact are biconditional.
  1. If you work hard only then you can succeed.
  2. You can go on leave only if your boss permits. The truth table for biconditional is as follows:
Truth table (p  qq  p)
p
q
p  q
q  p
T
T
T
T
T
F
F
F
F
T
F
T
F
F
T
T
Rule: p  q is true only when both p andq have the same truth value.




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