Relations and Functions Chapter 2 · Class 11 · NCERT Mathematics "Mathematics is the indispensable instrument of all physical research." — Berthelot

Cartesian Product Given two non-empty sets and , the Cartesian product is: Key properties: - If and , then - in general - - and

Cartesian Product — Example (NCERT Ex 3) Let , , Find and Both results are equal — this is always true.

Relations — Definition A relation from set to set is a subset of | Term | Meaning | |------|---------| | Domain | Set of all first elements (inputs) | | Range | Set of all second elements (outputs) used | | Codomain | The full set | > Note: Range Codomain If , we write — " is related to "

Relations — Example (NCERT Ex 7) Let , All pairs in R: - Domain - Codomain - Range Note: Domain because

Number of Relations The total number of relations from to : where and Why? Each subset of is a valid relation, and a set with elements has subsets. Example: , , so number of relations

Functions — Definition A function is a relation where every element of has exactly one image in The rule: one input exactly one output Requirements: - Every element of must be mapped - No element of can have two different images - Multiple elements of can map to the same element of

Function vs Not a Function (NCERT Ex 11) Given : R = {(2,1),(3,1),(4,2)} — Not a function Not every element of has an image R = {(2,2),(2,4),(3,3),(4,4)} — Not a function Element has two images: and R = {(1,2),(2,3),(3,4),(4,5),(5,6),(6,7)} — Function Every element has exactly one image

Special Functions (Part 1) Identity Function: , - Domain: , Range: Constant Function: , - Domain: , Range: Polynomial Function: where and is a non-negative integer

Special Functions (Part 2) Modulus Function: Domain: , Range: Signum Function: Domain: , Range:

Greatest Integer Function Examples: - - - - - Domain: , Range: Also called the floor function. The graph looks like a staircase.

Example — Squaring Function (NCERT Ex 13) , for | | | | | | | | | |-----|------|------|------|-----|-----|-----|-----| | | | | | | | | | - Domain - Range For , Domain , Range

Algebra of Functions For functions :

Algebra of Functions — Example (NCERT Ex 16) Given and

Common Mistakes Range vs Codomain: Range Codomain, but Range Codomain in general Cartesian Product: (order of pairs matters) Relations vs Functions: Every function is a relation, but not every relation is a function Function Rule: One element in domain cannot map to two different images (but two elements can map to the same image — that's allowed)

Summary | Concept | Definition | |---------|-----------| | Ordered pair | : first element , second | | Cartesian product | | | Relation | Any subset | | Domain | | | Range | | | Codomain | The full set | | Function | Relation where every has exactly one image | | Algebra of | , , defined pointwise |