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Equivalent
If  and  (i.e.,  , where  denotes implies), then  and  are said to be
equivalent, a relationship which is written symbolically in this work as  . The following
table summarizes some notations in common use.
| symbol | references |  | Moore
(1910, p. 150), Whitehead and Russell (1910, pp. 5-38), Carnap (1958, p. 8),
Curry (1977, p. 35), Itô (1986, p. 147), Gellert et al. 1989 (p. 333),
Cajori (1993, pp. 303 and 307), Church (1996, p. 78), Harris and Stocker
(1998, p. 471) |  | Wittgenstein
(1922, pp. 46-47), Cajori (1993, p. 313) |  | Mendelson
(1997, p. 13), Råde and Westergren 2004 (p. 9) |  | Harris
and Stocker (1998, back flap), DIN 1302 (1999) |  | Gellert et al. 1989
(p. 333), Harris and Stocker (1998, p. 471), Råde and Westergren
2004 (p. 9) |  | |
Equivalence is implemented in the Wolfram Language as Equal[A,
B, ...]. Binary equivalence has the following truth
table (Carnap 1958, p. 10), and is the same as  XNOR  , and  iff  .
Similarly, ternary equivalence has the following truth
table.
 |  |  |  | | T | T | T | T | | T | T | F | F | | T | F | T | F | | T | F | F | F | | F | T | T | F | | F | T | F | F | | F | F | T | F | | F | F | F | T |
The opposite of being equivalent is being nonequivalent.
Note that the symbol  is confusingly used in at least two other
different contexts. If  and  are "equivalent
by definition" (i.e.,  is defined
to be  ), this is written  , and " is congruent to  modulo  " is written
 .
SEE ALSO: Biconditional, Connective, Defined, Equivalence
Relation, Iff, Implies, Nonequivalent, XNOR
REFERENCES:
Cajori, F. A History of Mathematical Notations, Vol. 2. New York: Dover, p. 303,
1993.
Carnap, R. Introduction
to Symbolic Logic and Its Applications. New York: Dover, p. 8, 1958.
Church, A. Introduction to Mathematical Logic, Vol. 1. Princeton, NJ: Princeton University Press,
1996.
Curry, H. B. Foundations
of Mathematical Logic. New York: Dover, 1977.
Deutsches Institut für Normung E. V. DIN 1302: "General Mathematical Symbols and Concepts." Dec. 1, 1999.
Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR
Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold,
1989.
Harris, J. W. and Stocker, H. Handbook
of Mathematics and Computational Science. New York: Springer-Verlag, 1998.
Itô, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press,
1986.
Mendelson, E. Introduction
to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.
Moore, E. H. Introduction to a Form of General Analysis. New Haven, CT:
New Haven Math. Colloq., 1910.
Råde, L. and Westergren, B. Mathematics
Handbook for Science and Engineering. Berlin: Springer, 2004.
Whitehead, A. N. and Russell, B. Principia
Mathematica, Vol. 1. New York: Cambridge University Press, 1910.
Wittgenstein, L. Tractatus Logico-Philosophicus. London, 1922.
Referenced on Wolfram|Alpha: Equivalent
CITE THIS AS:
Weisstein, Eric W. "Equivalent." From
MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Equivalent.html
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