Mathematics Notes for Class 12 Relations And-Functions

Relations and Functions

Welcome to the chapter on Relations and Functions for Class 12. In this chapter, you will learn about relations, types of relations, functions, types of functions, and their properties. By the end of this chapter, you will be able to represent relations and functions, identify their types, and solve related problems.

Key Concepts

Relation: A relation from set A to set B is a subset of the Cartesian product A × B.
Function: A function is a special relation where every element of set A has exactly one image in set B.
Domain: The set of all first elements of the ordered pairs in a relation.
Range: The set of all second elements of the ordered pairs in a relation.
Codomain: The set B in a relation from A to B.

Types of Relations

Reflexive Relation: Every element is related to itself.
Symmetric Relation: If (a, b) is in the relation, then (b, a) is also in the relation.
Transitive Relation: If (a, b) and (b, c) are in the relation, then (a, c) is also in the relation.
Equivalence Relation: A relation that is reflexive, symmetric, and transitive.

Types of Functions

One-One (Injective): Every element of the domain maps to a unique element in the codomain.
Onto (Surjective): Every element of the codomain is the image of at least one element of the domain.
Bijective: Both one-one and onto.
Constant Function: Every element of the domain maps to the same element in the codomain.
Identity Function: Each element maps to itself.

Representation of Relations and Functions

Set-builder form
Roster form
Arrow diagrams
Graphical representation

Practice Questions

Let A = {1, 2, 3}, B = {4, 5}. List all possible relations from A to B.
Determine whether the relation R on set A = {1, 2, 3} defined by R = {(1,1), (2,2), (3,3), (1,2), (2,1)} is symmetric, reflexive, or transitive.
Give an example of a function that is one-one but not onto.
If f: R → R is defined by f(x) = 2x + 3, find f(0), f(1), and f(-2).
Explain the difference between domain, codomain, and range with examples.

Challenge Yourself

Prove that the relation "is parallel to" on the set of all lines in a plane is an equivalence relation.
Show that the function f: N → N defined by f(n) = n + 1 is one-one but not onto.
Draw the arrow diagram for the function f: {1, 2, 3} → {a, b, c} defined by f(1) = a, f(2) = b, f(3) = c.

Did You Know?

Every function is a relation, but not every relation is a function.
Functions are used in computer science, physics, and many real-life applications.

Glossary

Relation: A set of ordered pairs.
Function: A relation in which each input has exactly one output.
Domain: The set of all possible inputs.
Codomain: The set of all possible outputs.
Range: The set of actual outputs.

Answers to Practice Questions

There are 2⁶ = 64 possible relations (since A × B has 6 elements).
R is reflexive and symmetric, but not transitive.
f: N → N, f(x) = x + 1 is one-one but not onto.
f(0) = 3, f(1) = 5, f(-2) = -1
Domain: set of inputs; Codomain: set of possible outputs; Range: set of actual outputs. Example: f: {1,2,3} → {a,b,c}, f(1)=a, f(2)=b, f(3)=b. Domain = {1,2,3}, Codomain = {a,b,c}, Range = {a,b}.

Practice relations and functions to build a strong foundation in mathematics!

Mathematics Class 12 - Relations And-Functions Notes