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 SameQ[A,
B, ...]. Binary equivalence has the following truth
table (Carnap 1958, p. 10), and is the same as XNOR, and iff.
T
T
T
T
F
F
F
T
F
F
F
T
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 .
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.
XNOR 
, and 
iff 
.
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 
.
