Types of Relations in Class XII Mathematics

 Mathematics is often described as the study of patterns and logical connections. One of the most fundamental concepts that encapsulates this idea is the notion of a relation. In Class XII, the topic of Relations and Functions begins with the study of how two sets can be linked through ordered pairs. To understand functions deeply, one must first understand the types of relations that can exist between sets.

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1. Definition of Relation

Let $A$ and $B$ be two non-empty sets. A relation $R$ from $A$ to $B$ is a subset of the Cartesian product $A \times B$.

$$R \subseteq A \times B$$

If $(a, b) \in R$, we say that a is related to b under relation R.

Example:
Let $A = \{1,2,3\}, B = \{x,y\}$.
Possible relation: $R = \{(1,x), (2,y), (3,x)\}$.

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2. Types of Relations

When a relation is defined on a set, certain properties may or may not hold. These properties give rise to different types of relations.

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(a) Reflexive Relation

A relation $R$ on a set $A$ is reflexive if:

$$(a,a) \in R \quad \text{for all } a \in A$$

Example:
On $A = \{1,2,3\}$, the relation $R = \{(1,1), (2,2), (3,3)\}$ is reflexive.

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(b) Symmetric Relation

A relation $R$ on a set $A$ is symmetric if:

$$(a,b) \in R \implies (b,a) \in R$$

Example:
On $A = \{1,2\}$, the relation $R = \{(1,2), (2,1)\}$ is symmetric.

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(c) Transitive Relation

A relation $R$ on a set $A$ is transitive if:

$$(a,b) \in R \ \text{and} \ (b,c) \in R \implies (a,c) \in R$$

Example:
On $A = \{1,2,3\}$, if $R = \{(1,2), (2,3), (1,3)\}$, then $R$ is transitive.

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(d) Equivalence Relation

A relation $R$ on a set $A$ is called an equivalence relation if it is:

1. Reflexive,

2. Symmetrical, and

3. Transitive.

Example:
On the set of integers $\mathbb{Z}$, the relation $a \sim b \iff a-b$ is divisible by 5 is an equivalence relation.

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(e) Universal Relation

A relation $R$ on a set $A$ is called universal if:

$$R = A \times A$$

That is, every element of $A$ is related to every element of $A$.

Example:
For $A = \{1,2\}$, $R = \{(1,1), (1,2), (2,1), (2,2)\}$.

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(f) Empty Relation

A relation $R$ on a set $A$ is called empty if:

$$R = \varnothing$$

That is, no element of $A$ is related to any element of $A$.

Example:
On $A = \{1,2,3\}$, the relation $R = \varnothing$ is an empty relation.

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(g) Identity Relation

A relation $R$ on a set $A$ is called an identity relation if:

$$R = \{(a,a): a \in A\}$$

That is, every element is related only to itself.

Example:
On $A = \{1,2,3\}$, $R = \{(1,1), (2,2), (3,3)\}$.

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3. Summary Table

Type of Relation	Definition	Example
Reflexive	$(a,a) \in R$ for all $a$	$(1,1), (2,2), (3,3)$
Symmetric	$(a,b) \in R \Rightarrow (b,a) \in R$	$(1,2), (2,1)$
Transitive	$(a,b),(b,c) \in R \Rightarrow (a,c) \in R$	$(1,2),(2,3) \Rightarrow (1,3)$
Equivalence	Reflexive + Symmetric + Transitive	$a \sim b \iff a-b \equiv 0 \pmod{5}$
Universal	$R = A \times A$	$(1,1),(1,2),(2,1),(2,2)$
Empty	$R = \varnothing$	None
Identity	$R = \{(a,a): a \in A\}$	$(1,1),(2,2)$

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4. Conclusion

The classification of relations provides a logical framework for understanding connections within sets. Reflexive, symmetric, and transitive relations form the basis of equivalence relations, while identity, empty, and universal relations represent boundary cases. For Class XII learners, mastering these types not only builds a strong foundation for functions and mappings but also lays the groundwork for higher studies in abstract algebra, computer science, and discrete mathematics.