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If and only if
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==In terms of Euler diagrams== <gallery widths="270"> File:Example of A is a proper subset of B.svg|''A'' is a proper subset of ''B''. A number is in ''A'' only if it is in ''B''; a number is in ''B'' if it is in ''A''. File:Example of C is no proper subset of B.svg|''C'' is a subset but not a proper subset of ''B''. A number is in ''B'' if and only if it is in ''C'', and a number is in ''C'' if and only if it is in ''B''. </gallery> [[Euler diagram]]s show logical relationships among events, properties, and so forth. "P only if Q", "if P then Q", and "PβQ" all mean that P is a [[subset]], either proper or improper, of Q. "P if Q", "if Q then P", and QβP all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other.
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