Types of Relations
2026-02-28 17:36 Diff
https://brightchamps.com/en-us/math/algebra/types-of-relations

Empty Relation

Definition: In empty relations, no elements are related.
Example: For student set A, the relation “is taller than” is empty if all students are the same height: R =  or R = {}.

Universal Relation

Definition: Every pair of elements is related
Example: For set A = {1, 2}, R = {(1, 1), (1, 2), (2, 1), (2, 2)} = A × A

Identity Relation

Definition: A relation in which every element in a set is paired with itself only.
Example: For A = {a, b, c}, R = {(a, a), (b, b), (c, c)}

Reflexive Relation

Definition: Every element in the set is related to itself. It can also include other pairs, but the key requirement is that each element must relate to itself, such as (a, a), (b, b), etc.
Example: On A = {1, 2}, R = {(1, 1), (2, 2),  (1, 2)} 

Symmetric Relation

Definition: If a relates to b, then b relates to a
Example: R = {(2, 3), (3, 2) on  (2, 3)} 

Transitive Relation

Definition: A relation is called transitive if, whenever an element a is related to b, and b is related to c, then a must also be related to c.
Example: For a set A = {1, 2, 3}, the transitive relation is given as R = {(1, 2), (2, 3), (1, 3)} 

Equivalence Relation

Definition: An equivalence relation is reflexive, symmetric, and transitive altogether.
Example: On the set of real numbers R, the relation “is equal to” (=) is an equivalence relation because:
Every number equals itself (reflexive)
If a = b, then b = a (symmetric)
If a = b and b = c, then a = c (transitive)

Antisymmetric Relation

Definition: A relation is antisymmetric if, whenever (a, b) and (b, a) are both in the relation, then it must be that a = b. 
Example: Consider the relation "≤" (less than or equal to) on numbers.
If a ≤ b and b ≤ a, then the only way both can be true is if a = b.
So, "≤" is an antisymmetric relation.

Inverse Relation

Definition: The inverse of a relation is formed by reversing the order of each pair in the original relation.
Example: If R = {(1, 2), (3, 5)}, then the inverse relation R-1 = {(2, 1), (5, 3)}