equality (original) (raw)

In fact, for any set S, the smallest equivalence relation on S is the equality (by smallest we that it is contained in every equivalence relation on S). This offers a definition of “equality”. From this, it is clear that there is only one equality relation on S. Its equivalence classes are all singletons {x} where x∈S.

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 S 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 S 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. (x+y)2=x2+y2 is an identity in a field with characteristic (http://planetmath.org/Characteristic) 2.