Relations And Functions: Complete Notes For CUET 2027 by PW

Relations and functions involve several definitions, properties and classification rules that can be confusing during revision. Understanding how these concepts differ and how to apply them to ordered pairs, sets and graphs is essential for solving CUET Mathematics questions.

PW’s one-shot revision lectures on Relations & Functions help you revisit the chapter’s key concepts in one session. Watch the lecture alongside these notes, revise important examples, and practise previous year questions (PYQs) to reinforce your understanding.

Meaning And Representation Of A Relation

A relation describes a connection between elements according to a specified condition.

Let A = {1, 2, 3, 4, 5}. If two numbers are related when their sum is 6, the relation is:

R = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

Every ordered pair satisfies the given condition. In general, when a set and a condition are provided, all pairs satisfying that condition form the relation.

Also Check: PW CUET Wallah YouTube Channel

Domain, Range, And Codomain

For a relation represented by ordered pairs:

R = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

Here, the domain is {1, 2, 3, 4, 5}, and the range is {5, 4, 3, 2, 1}. If the relation is defined from set A to set B, then B is its codomain.

Total Number Of Relations

If a set A has n elements, the total number of relations from A to A is 2 raised to the power n squared.

For A = {1, 2, 3, 4, 5}, n = 5. Therefore, the total number of relations is 2 raised to the power 25.

Types Of Relations

Empty, Void, Or Null Relation

An empty relation, also called a void relation, contains no ordered pair. It occurs when no elements satisfy the given condition. It is denoted by the empty set symbol.

Universal Relation

A universal relation relates every element of A to every element of A. It is equal to the Cartesian product A times A.

Identity Relation

The identity relation on A contains only self-pairs. For A = {1, 2, 3, 4}:

I = {(1, 1), (2, 2), (3, 3), (4, 4)}

Every element must be related to itself, and no additional pair can be present. (*Memory Tip: Think of the identity relation as a relation in which every element is related to itself and only itself.)*

The identity relation is the smallest equivalence relation.

Reflexive Relation

A relation is reflexive if every element is related to itself. Thus, all pairs of the form (a, a) must be present.

Additional pairs may also be included.

For a set with n elements, the number of reflexive relations is 2 raised to the power n squared minus n.

Symmetric Relation

A relation is symmetric if the presence of (a, b) requires the presence of (b, a).

For example, if (2, 3) belongs to the relation, then (3, 2) must also belong to it. Self-pairs are automatically symmetric. (*Memory Tip: A symmetric relation requires that every ordered pair must have its reverse pair.)*

Transitive Relation

A relation is transitive if (a, b) and (b, c) together imply (a, c).

To test transitivity, check every possible chain of ordered pairs. If (2, 3) and (3, 1) are present but (2, 1) is missing, the relation is not transitive. A single failure makes the entire relation non-transitive.

If (a, b) is present but no pair begins with b, there is no further condition to check.

Equivalence Relation

A relation is an equivalence relation when it is:

1. Reflexive

2. Symmetric

3. Transitive

The number of equivalence relations follows the Bell number sequence. For sets containing one, two, three, and four elements, the values are 1, 2, 5, and 15. (*Memory Tip: Remember the first four values as 1, 2, 5, 15.)*

Function As A Special Relation

A function is a special relation from set A to set B. It assigns exactly one image in B to every element of A.

For a function written as f from A to B:

• Every element of A must have an image.

• No element of A can have two different images.

If (1, 2) is an ordered pair, then 2 is the image of 1, and 1 is the preimage of 2.

(*Memory Tip: One input cannot have two outputs, but different inputs may have the same output.)*

For f from A to B, A is the domain, B is the codomain, and the range contains only the codomain elements that actually occur as images.

Function Tests Using Ordered Pairs And Graphs

A relation is not a function if a first element is repeated with different second elements. For example, (4, 4) and (4, 5) show that 4 has two images.

However, repeated second elements are allowed. In (5, 12) and (6, 12), two different inputs have the same image, so the relation can still be a function.

The Vertical Line Test checks whether a graph represents a function. If any vertical line cuts the graph at more than one point, the graph is not a function.

One-One And Many-One Functions

A function is one-one if different domain elements have different images. In symbols, equal images must come from equal inputs.

A function is many-one when two or more different domain elements have the same image.

The modulus function f(x) = absolute value of x is not one-one because positive and negative values have the same image. Similarly, functions involving even powers, such as x squared plus 3, are generally not one-one over their full real domain.

The Horizontal Line Test checks whether a function is one-one. If any horizontal line intersects the graph at more than one point, the function is not one-one. This applies to graphs such as y = x squared, sine x, and cosine x over their usual domains.

A derivative that remains positive or remains negative over the relevant domain also indicates a one-one function.

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Onto, Into, And Bijective Functions

A function is onto when every codomain element has at least one preimage. Therefore:

Range = Codomain

If the range is not equal to the codomain, the function is into.

For f(x) = x squared from the real numbers to the real numbers, the range is from zero to infinity, while the codomain is all real numbers. Hence, the function is not onto.

A function is bijective if it is both one-one and onto. Bijective functions can exist only when the two sets have the same number of elements. If each set has m elements, the number of bijective functions is m factorial.

For a one-one function from a set with n elements to a set with m elements, n must be less than or equal to m. The number of such functions is the number of ways to select distinct images for all domain elements.

For an onto function, the domain must have at least as many elements as the codomain.

Application: Classifying 10 Minus X Squared

Consider f(x) = 10 minus x squared.

f(-2) and f(2) are both equal to 6. Thus, two different inputs have the same image, so the function is not one-one.

Also, x squared is always non-negative. Therefore, 10 minus x squared is always less than or equal to 10. Its range is all values up to 10, while its codomain is all real numbers. Since the range is not the codomain, the function is not onto.

Hence, the function is neither one-to-one nor onto, and therefore it is not bijective.

Watch PW’s one-shot revision lecture on Relations and Functions for CUET 2027 to revisit key concepts and examples. Use it alongside these notes and practise PYQs to strengthen your preparation.

Relations and Functions become easier to revise when you understand the logic behind each property instead of memorising definitions in isolation. Use these notes to revisit confusing concepts, watch PW’s one-shot lecture for another round of revision, and solve CUET PYQs to apply what you have learned.