### Equivalence

Definition:

- Two propositions connected by 'if and only if'.

Example:

- "I will give you a quarter if and only if I give you a penny."

  Following the logical convention of having a single letter represent a simple atomic proposition:

  - P = "I will give you a penny."
  - Q = "I will give you a Quarter."

  And replacing 'if and only if' with the symbol for equivalence: ' = ' The above equivalence can now be symbolized:

  - (Q = P)

  Equivalencies are false when the components on either side of the '=' do not have the same value: when one is true and the other is false. And if the components on either side of '=' have the same value (both are true or both are false) , then the equivalence is true. The truth value of an equivalence is thus a function of the truth value of its components. In logic this is characterized by a truth table.

  EQUIVALENCE

  INPUT | OUTPUT
  --- | ---
  P......Q | P = Q
  T......T | T
  T......F | F
  F......T | F
  F......F | T

  Examples of equivalencies:
  (C = ~W)
  [(B > N) = (U . V)]