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 mean 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: \begin{eqnarray*} 1 + 1 & = & 2 \\ e^{i \pi} & = & -1 \\ \mathbb{R}[i] & = & \mathbb{C} \end{eqnarray*}(The second example is Euler's identity.)

Remark 1. Although the four characterising properties, reflexivity, symmetry, transitivity and antisymmetry, determine the equality on $S$ uniquely, they cannot be thought to form the definition of the equality, since the concept of antisymmetry already contains 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 $2$