equality

In any set $S$, the equality, denoted by “$=$”, is a binary relation which is reflexive, symmetric, transitive and antisymmetric, i.e. it is an antisymmetric equivalence relation on $S$, or which is the same thing, the equality is a symmetric partial order on $S$.

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  $x\in S$.

The concept of equality is essential in almost all branches of mathematics. A few examples will suffice:

$1+1$$=$$2$

$e^{i\pi }$$=$$-1$

$ℝ[i]$$=$$ℂ$

(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.  ${(x+y)}^2=x^2+y^2$  is an identity in a field with characteristic (http://planetmath.org/Characteristic) $2$.