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∈S.
The concept of equality is essential in almost all branches of mathematics. A few examples will suffice:
1+1 | = | 2 | ||
eiπ | = | -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=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 |