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102 LearnersLast updated on September 30, 2025
Types of Relations
Relations describe how elements from one set are connected to elements in another set. The types of relations include universal, symmetric, reflexive, transitive, and so on. If a relation is at the same time reflexive, symmetric, and transitive, then it is an equivalence relation.
What is a Relation?
A relation is a set of ordered pairs that connects elements from a domain to elements in a codomain. A relation can be defined by a rule, such as y = x2, which pairs each value of x from set A with a corresponding y value in set B.
What are the 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)}
Sets and Relations
Relations are built using elements from sets. A relation shows how elements from two sets connect. When you have two sets and need to see if any elements from either of those sets match or link, you use a relation. For example, an empty relation means that no elements from the two sets are connected—there are no matching pairs at all.
Key Formula for Relations and Their Types
- Empty Relation: R=AA
- Universal Relation: R=AA
- Identity Relation: I={(a,a):aA}
- Reflexive Relation: (a,a)R for all aA
- Symmetric Relation: aRbbRa,a,ba
- Transitive Relation: aRb and bRcaRc a, b, c A
- Inverse Relation: R-1={(b,a):(a,b)R
How to Determine Relation Types?
Relations connect elements within a set or between two different sets. To determine a relation type, check three key properties: reflexive, where every element relates to itself, and symmetric. If a relates to b, then b relates to a, and transitive, where if a relates to b and b to c, then a relates to c. When a relation is reflexive, symmetric, and transitive, it is called an equivalence relation.
How to Represent a Relation?
Roster (List) Form
Example: R = {(3, 9), (4, 16), (5, 25)}
Meaning: We list all related pairs between P and Q.
Arrow Diagram
Format: Draw arrows from each element in P to its corresponding square in Q
39
416
525
Meaning: A visual map showing who connects to whom.
Real-Life Applications of the Types of Relations
Types of relations have many real-life applications. They are used in social networks, hierarchies, databases, and algorithms. Some of these applications are mentioned below:
- Reflexive Relation: Every person relating to themselves represents the idea of reflexivity in a relation; every user has the right to a unique user profile, which is used by database designers and security engineers to make sure self-access naturally exists.
- Symmetric Relation: “Is married to” relationship. If A is married to B, then B is married to A. This is used by social network platforms and matchmaking systems to model mutual relationships.
- Transitive Relations: In organizational hierarchy, if manager A supervises B, and B supervises C, then A indirectly supervises C. In HR systems or network routing, transitive relations help identify indirect connections by linking paths through multiple steps.
- Equivalence relation: Equivalence relations are used to group items that share a common property. For example, people can be grouped by the same last name or color. In computing, equivalence relations help in partitioning data into equivalent classes, such as in modular arithmetic, database classification, or simplifying lookups by grouping similar entries.
- Inverse relation: If relation R shows which students borrow which books, then the inverse relation R-1 shows which books are borrowed by which students. Used by library systems, function, or query reversals to support reverse lookup and swapped viewpoints.
Common Mistakes in Types of Relations and How to Avoid Them
Students make mistakes while working with relational types. They might incorrectly assume that all relations are symmetric or transitive, which can lead to wrong answers. To avoid this, here are some common mistakes and ways to avoid them.
Mistake 1
Missing reflexive pairs
Assuming a relation is reflexive without including each (a, a), recheck every aA has (a, a) in R. If A = {1, 2, 3} but (3, 3) is missing, R is not reflexive.
Mistake 2
Mixing up symmetric and transitive
Thinking symmetry implies transitivity, treat each property separately—for example, a relation can be symmetric but not transitive.
Mistake 3
Assuming symmetric and transitive implies reflexive
Thinking symmetric and transitive guarantees reflexivity. Always verify reflexive pairs explicitly.
Mistake 4
Overlooking empty-relation edge cases
Students often forget that an empty relation (∅) is always symmetric and transitive, because there are no counterexamples. However, it is not reflexive—unless the set itself is empty—because it does not contain any (a, a) pairs.
Mistake 5
Ignoring domain or set details
Always define the set clearly before checking relation properties, because the elements in the set determine whether the relation is reflexive, symmetric, or transitive.
Solved Examples
Problem 1
Let A = {1, 2, 3} Let R = {(1, 1), (2, 2), (3, 3)}
Reflexive, symmetric, transitive.
Explanation
Each element is related to itself → Reflexive
For every (a, a), the reverse (a, a) also exists → Symmetric
No chain violations; all possible (a, a) hold → Transitive
Problem 2
Let A = {1, 2, 3} Let R = {(1, 2), (1, 3), (2, 3)}
Transitive only
Explanation
Not reflexive: missing (1,1), (2,2), (3,3)
Not symmetric: (1, 2) exists but (2, 1) doesn’t
Transitive: (1, 2) and (2, 3) imply (1, 3), which is included
Problem 3
Relation R on real numbers R, where xRy if x y
Transitive only
Explanation
Not reflexive: no number is less than itself
Not symmetric: if x < y, then y is not < x
Transitive: if x < y and y < z, then x < z
Problem 4
Let R be congruence mod 3 on integers Z: xRy if x y (mod 3)
Reflexive, symmetric, transitive equivalence relation
Explanation
x ≡ x mod 3 → Reflexive
If x ≡ y mod 3, then y ≡ x mod 3 → Symmetric
If x ≡ y and y ≡ z mod 3, then x ≡ z → Transitive
Problem 5
Let A = {1, 2, 3, 4} Relation R = {(x, y) | xy}
Reflexive, antisymmetric, transitivepartial order.
Explanation
x ≤ x for all x → Reflexive
If x ≤ y and y ≤ x, then x = y → Antisymmetric
If x ≤ y and y ≤ z, then x ≤ z → Transitive
FAQs on Types of Relations
1.How is a relation different from a function?
2.What is an empty relation?
3.What does it mean for a relation to be reflexive?
4.When is a relation symmetric?
5.What does transitive mean in a relation?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How do technology and digital tools in United States support learning Algebra and Types of Relations?
8.How can cultural or local activities in United States support learning Algebra topics such as Types of Relations?
9.Does learning Algebra support future career opportunities for students in United States?
