Understanding Relations and Its Types: Definition, Representation and Examples

One of the types of relations in Math is empty relation. A relation R on a set A will be an empty relation, if no element of A is related to any element of A. Therefore, $\varphi \subset$ A $\times$ A is called a void or an empty relation on A. Class 12th board exams ask questions related to this topic. Students can go through NCERT exercises for understanding the types of questions that will be asked in the examination.

Example: Let us say we have two sets: Set A: $\left\{1, 2, 3\right\}$ Set B: $\left\{a, b, c\right\}$

A relation R could be defined as: R = $\left\{\left(x, y\right) | x + y = 10\right\}$

In this case, there are no elements in either set A or set B that, when summed, equal 10.

Therefore, the relation R would be an empty relation; R = $\varphi$

A relation R on a set A is said to be a universal relation, if each element of A is related to every element of A. Therefore, A $\times$ A $\subset$ A $\times$ A is called a universal relation on A. Let us know learn about this type of relation through an example.

Example: A: $\left\{1, 2, 3\right\}$

The Cartesian product A A would be: $\left\{\begin{array}{l}\left(1, 1\right), \left(1, 2\right), \left(1, 3\right), \left(2, 1\right), \left(2, 2\right), \\ \left(2, 3\right), \left(3, 1\right), \left(3, 2\right), \left(3, 3\right)\end{array}\right\}$

If R = A $\times$ A, then R would be the universal relation on A. It means that every element in A is related to every other element in A.

Some of the relevant reads for you:

NCERT solutions	NCERT Class 11 Math Notes
NCERT Class 12 Maths Notes for CBSE	CBSE Class 12 NCERT Notes

Let A be a non - empty set. Then, a relation I_A on A is called an identity relation if and only if the image of a $\in$ A on I_A is a $\forall$ a $\in$ A

i.e., I_A = $\left\{\left(a,a\right):a\in A\right\}$

Example: If A: $\left\{1, 2, 3\right\}$ , the identity relation would be $\left\{\left(1, 1\right), \left(2, 2\right), \left(3, 3\right)\right\}$ .

Every identity relation is a reflexive relation as it includes all ordered pairs where the first and second element are the same. However, not all reflexive relations are identity relations (e.g.,) is reflexive but not identity because it also includes (1,2). NEET and JEE Main aspirants must practice questions related to this concept since this chapter is included in both class 11th and 12th. 

Also known as converse relation; an inverse relation is formed by reversing the direction of all ordered pairs in the original relation. If R is a relation from set A to set B, then its inverse relation R⁻¹ is a relation from set B to set A. Questions around inverse relation are also asked in CUET and IISER exams which is why student must practice question related to this type of relation.
Let R = $\left\{\left(a,b\right):a,b\in A\right\}$ be a relation on A. Then a relation $\left\{\left(b,a\right):a,b\in A\right\}$ is called an inverse of relation R and is denoted by R^-1.

$\therefore$ R^-1 = $\left\{\left(b,a\right):b\in B,a\in A,\left(a,b\right)\in R\right\}$

Example: Original Relation: Let R = $\left\{\left(1, 2\right), \left(3, 4\right), \left(5, 6\right)\right\}$ .

Inverse Relation: The inverse, R^-1, would be R^-1 = $\left\{\left(2, 1\right), \left(4, 3\right), \left(6, 5\right)\right\}$ .

If you have a mapping from a set A to a set B (e.g., R: A $\to$ B),

the inverse relation maps from B to A (R^-1: B $\to$ A).

One of the other types of relations is reflexive relation. A relation R on a set A is said to be reflexive, if every element of A is related to itself. Let us consider an example to understand this:

Equality: Every element is equal to itself (e.g., a = a)

Greater than or equal to: Every element is greater than or equal to itself (e.g., a $\ge$ a).

Less than or equal to: Every element is less than or equal to itself (e.g., a $\le$ a).

Divisibility: Every number divides itself.

Subset: Every set is a subset of itself.

Non - Examples:

Being older than: If X is older than Y, then Y is not necessarily older than X.

Being parent of: If X is the parent of Y, then Y is not the parent of X.

Is less than: If a < b, then b < a is not true.

A relation R on a set A is said to be transitive, if (a, b) $\in$ R and (b, c) $\in$ R $\Rightarrow$ (a, c) $\in$ R $\forall$ a, b, c $\in$ A. Let's look at an example to understand this relation type:

Examples:

Is less than (<): If x < y and y < z, then x < z.

Is equal than (=): If x = y and y = z, then x = z.

Is divisible by: If a is divisible by b and b is divisible by c, then a is divisible by c.

Is an ancestor of: If person A is an ancestor of person B, and person B is an ancestor of person C, then A is an ancestor of C.

Is the parent of: If person A is the parent of person B, and person B is the parent of person C, it doesn’t mean A is the parent of C (A would be a grandparent of C).

This is the last type of relations in math that we have discussed here. Similarly, there are different types of functions that are covered as a separate topic. A relation R on a set A is said to be equivalence relation if and only if R is reflexive, symmetric and transitive. Below is an example that explanins equivalence relation:

Examples:

Is equal to (=): This is the most fundamental example. If you have 2 = 2, 2 = 3, 3 = 4,then 2 = 4.

Is similar to ( $\sim$ ): Two triangles are similar if they have the same shape but possibly different sizes.

Has the same birthday: If you have two people who share the same birthday, that’s an equivalence relation.

Is congruent to: Two geometric shapes are congruent if they have the same size and shape.

Congruence modulo n ( $\equiv$ ): For integers, a $\equiv$ b (mod n) if their difference is divisible by n.

For example, 7 $\equiv$ 2 (mod 5) because 7 – 2 = 5, which is divisible by 5.