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Relations

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Last modified: 2026-11-06 15:35:15 (PST)

Remark. A relation says which pairs of elements of a set are related. This page defines the properties that make a relation behave like equality, and the groups of elements that such a relation puts together.

1 Properties of relations

Definition 1 (Reflexive relation) Let \(\sim\) be a relation on a set \(S\). The relation \(\sim\) is reflexive (Wikipedia contributors 2026c) if \(a \sim a\) for every \(a \in S\).

Example 1 (Reflexive and non-reflexive relations on the real numbers) On \(\mathbb{R}\), the relation \(=\) is reflexive, since \(a = a\) for every real number \(a\). The relation \(\le\) is also reflexive, since \(a \le a\) for every real number \(a\). The relation \(<\) is not reflexive, since \(2 < 2\) is false.

Definition 2 (Symmetric relation) Let \(\sim\) be a relation on a set \(S\). The relation \(\sim\) is symmetric (Wikipedia contributors 2026d) if, for all \(a, b \in S\), \(a \sim b\) implies \(b \sim a\).

Example 2 (Symmetric and non-symmetric relations on the real numbers) On \(\mathbb{R}\), the relation \(=\) is symmetric: whenever \(a = b\), also \(b = a\). The relation \(\le\) is not symmetric, since \(2 \le 5\) is true but \(5 \le 2\) is false.

2 Equivalence relations

Definition 3 (Equivalence relation) Let \(\sim\) be a relation on a set \(S\). The relation \(\sim\) is an equivalence relation (Wikipedia contributors 2026b) if it is reflexive, symmetric, and transitive.

Example 3 (Integers with an even difference) On the integers, define the relation \(\sim\) by \(a \sim b\) when \(a - b\) is even. For example, \(7 \sim 3\), since \(7 - 3 = 4 = 2 \cdot 2\), but \(7 \not\sim 4\), since \(7 - 4 = 3\) is odd. The relation \(\sim\) is an equivalence relation.

Reflexive. For every integer \(a\), \(a - a = 0 = 2 \cdot 0\), so \(a \sim a\).

Symmetric. Suppose \(a \sim b\), so \(a - b = 2k\) for some integer \(k\). Then

\[ \begin{aligned} b - a &= -(a - b) && \text{(negate the difference)} \\ &= -2k && \text{(} a - b = 2k \text{)} \\ &= 2(-k), && \text{(factor out } 2 \text{)} \end{aligned} \]

and \(-k\) is an integer, so \(b \sim a\).

Transitive. Suppose \(a \sim b\) and \(b \sim c\), so \(a - b = 2j\) and \(b - c = 2k\) for some integers \(j\) and \(k\). Then

\[ \begin{aligned} a - c &= (a - b) + (b - c) && \text{(the } b \text{ terms cancel)} \\ &= 2j + 2k && \text{(} a - b = 2j \text{ and } b - c = 2k \text{)} \\ &= 2(j + k), && \text{(distributive law)} \end{aligned} \]

and \(j + k\) is an integer, so \(a \sim c\).

Definition 4 (Equivalence class) Let \(\sim\) be an equivalence relation on a set \(S\), and let \(a \in S\). The equivalence class (Wikipedia contributors 2026a) of \(a\) is the set of elements of \(S\) related to \(a\):

\[ [a] \stackrel{\text{def}}{=}{\mathopen{}\left\{b \in S : b \sim a\right\}\mathclose{}}. \]

Example 4 (Equivalence classes of integers with an even difference) Let \(\sim\) be the equivalence relation of Example 3 on the integers. The equivalence class \([0]\) holds the integers \(b\) with \(b - 0\) even, which are the even integers: \([0] = \mathopen{}\left\{\ldots, -4, -2, 0, 2, 4, \ldots\right\}\mathclose{}\). The class \([1]\) holds the integers \(b\) with \(b - 1\) even, which are the odd integers: \([1] = \mathopen{}\left\{\ldots, -3, -1, 1, 3, 5, \ldots\right\}\mathclose{}\). The class \([2]\) is the same set as \([0]\), since \(2 \sim 0\).

Definitions on this page

References

Wikipedia contributors. 2026a. Equivalence Class — Wikipedia, the Free Encyclopedia. https://en.wikipedia.org/wiki/Equivalence_class.
Wikipedia contributors. 2026b. Equivalence Relation — Wikipedia, the Free Encyclopedia. https://en.wikipedia.org/wiki/Equivalence_relation.
Wikipedia contributors. 2026c. Reflexive Relation — Wikipedia, the Free Encyclopedia. https://en.wikipedia.org/wiki/Reflexive_relation.
Wikipedia contributors. 2026d. Symmetric Relation — Wikipedia, the Free Encyclopedia. https://en.wikipedia.org/wiki/Symmetric_relation.
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