# 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:

$1+1$ | $=$ | $2$ | ||

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

$\mathbb{R}[i]$ | $=$ | $\u2102$ |

(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) $2$.

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 |