In any set , the equality, denoted by “”, is a binary relation which is reflexive, symmetric, transitive and antisymmetric, i.e. it is an antisymmetric equivalence relation on , or which is the same thing, the equality is a symmetric partial order on .
In fact, for any set , the smallest equivalence relation on is the equality (by smallest we that it is contained in every equivalence relation on ). This offers a definition of “equality”. From this, it is clear that there is only one equality relation on . Its equivalence classes are all singletons where .
The concept of equality is essential in almost all branches of mathematics. A few examples will suffice:
(The second example is Euler’s identity.)
Remark 1. Although the four characterising , reflexivity, symmetry (http://planetmath.org/Symmetric), transitivity and antisymmetry (http://planetmath.org/Antisymmetric), determine the equality on uniquely, they cannot be thought to form the definition of the equality, since the concept of antisymmetry already the equality.
Remark 2. An equality (equation) in a set may be true regardless to the values of the variables involved in the equality; then one speaks of an identity or identic equation in this set. E.g. is an identity in a field with characteristic (http://planetmath.org/Characteristic) .
|Date of creation||2013-03-22 18:01:26|
|Last modified on||2013-03-22 18:01:26|
|Last modified by||pahio (2872)|