Properties of Relations in Set Theory

Sets are defined as a collection of well-defined objects. Relation refers to a relationship between the elements of 2 sets A and B. It is represented by R. We say that R is a relation from A to A, then R ⊆ A×A. A relation from set A to set B is a subset of A×B. i.e aRb ↔ (a,b) ⊆ R ↔ R(a, b). In this article, we will learn the important properties of relations in set theory. Read More.

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9 Important Properties Of Relations In Set Theory

1. Identity Relation: Every element is related to itself in an identity relation. It is denoted as I = {(a, a), a ∈ A}.

2. Empty relation: There will be no relation between the elements of the set in an empty relation. It is the subset ∅.

3. Reflexive relation: Every element gets mapped to itself in a reflexive relation. A relation R in a set A is reflexive if (a, a) ∈ R for all a∈R.

4. Irreflexive relation: If any element is not related to itself, then it is an irreflexive relation.

5. Inverse relation: When a set has elements which are inverse pairs of another set, then the relation is an inverse relation. For example, if A = {(p,q), (r,s)}, then R-1 = {(q,p), (s,r)}. Inverse relation is denoted by R-1 = {(b, a): (a, b) ∈ R}.

6. Symmetric relation: A relation R is symmetric a symmetric relation if (b, a) ∈ R is true when (a,b) ∈ R. For example R = {(3, 4), (4, 3)} for a set A = {3, 4}. Symmetric relation is denoted by

aRb ⇒ bRa, ∀ a, b ∈ A.

7. Transitive relation: A relation is transitive, if (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R. It is denoted by aRb and bRc ⇒ aRc ∀ a, b, c ∈ A

8. Equivalence relation: A relation is called equivalence relation if it is reflexive, symmetric, and transitive at the same time.

9. Universal relation: A relation is said to be universal relation, If each element of A is related to every element of A, i.e. R = A × A.

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