1.1 Introduction
In mathematics, relations and functions help us understand how elements of one set are connected to elements of another set. These concepts are widely used in algebra, calculus, and real-life problem solving.
- A set is a well-defined collection of objects.
- When elements of one set are connected with elements of another set, a relation is formed.
- A function is a special type of relation with specific rules.
This chapter builds the foundation for many advanced topics in Class 12 mathematics.
1.2 Types of Relations
Let A and B be two non-empty sets. A relation R from set A to set B is a subset of A × B.
🔹 1. Empty Relation
A relation that has no elements.
Example: If A = {1, 2}, then R = ∅ is an empty relation.
🔹 2. Universal Relation
A relation that contains all possible ordered pairs from A × B.
🔹 3. Reflexive Relation
A relation R on a set A is reflexive if (a, a) ∈ R for every a ∈ A.
🔹 4. Symmetric Relation
A relation R is symmetric if (a, b) ∈ R implies (b, a) ∈ R.
🔹 5. Transitive Relation
A relation R is transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R.
🔹 6. Equivalence Relation
A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
1.3 Types of Functions
A function is a relation where each element of the domain is mapped to exactly one element of the codomain.
🔹 1. One-One (Injective) Function
Different elements of the domain have different images in the codomain.
🔹 2. Many-One Function
Two or more elements of the domain are mapped to the same element of the codomain.
🔹 3. Onto (Surjective) Function
Every element of the codomain has at least one pre-image in the domain.
🔹 4. Into Function
At least one element of the codomain has no pre-image in the domain.
🔹 5. One-One and Onto (Bijective) Function
A function that is both one-one and onto.
Only bijective functions have inverse functions.
1.4 Composition of Functions and Invertible Function
🔹 Composition of Functions
If f: A → B and g: B → C, then the composition g ∘ f is defined as:(g∘f)(x)=g(f(x))
- The composition is possible only when the range of f is a subset of the domain of g.
- Composition of functions is associative, but generally not commutative.
🔹 Invertible Function
A function is said to be invertible if its inverse exists.
- Only bijective functions are invertible.
- If f⁻¹ is the inverse of function f, then: f(f−1(x))=xandf−1(f(x))=x