In any set S, the equality, denoted by “=”, is a binary relationMathworldPlanetmath which is reflexiveMathworldPlanetmathPlanetmath, symmetric, transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath and antisymmetric, i.e. it is an antisymmetric equivalence relationMathworldPlanetmath on S, or which is the same thing, the equality is a symmetric partial orderMathworldPlanetmath 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 classesMathworldPlanetmath are all singletons {x} where  xS.

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

1+1 = 2
eiπ = -1
[i] =

(The second example is Euler’s identityPlanetmathPlanetmathPlanetmath.)

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.

Title equality
Canonical name Equality
Date of creation 2013-03-22 18:01:26
Last modified on 2013-03-22 18:01:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Topic
Classification msc 06-00
Related topic Equation
Defines equality relation
Defines identity
Defines identic equation