Relations and Functions Notes: Class 12 Mathematics Chapter 1
Class 12 Mathematics Chapter 1 Relations and Functions covers different types of relations (reflexive, symmetric, transitive, equivalence relations), types of functions (one-one, onto, bijective), and the composition and invertibility of functions.
Overview of Topics & Subtopics (Click to Jump)
1. Types of Relations
Short Note:
• A relation R on a set A is reflexive if (a, a) ∈ R for every a ∈ A; symmetric if (a, b) ∈ R implies (b, a) ∈ R; and transitive if (a, b), (b, c) ∈ R implies (a, c) ∈ R.
• An equivalence relation satisfies all three properties — reflexive, symmetric, and transitive — simultaneously, and it partitions the set into disjoint equivalence classes.
2. Types of Functions
Short Note:
• A function f: A → B is one-one (injective) if distinct elements of A always map to distinct elements of B; it is onto (surjective) if every element of B has at least one preimage in A.
• A function that is both one-one and onto is called bijective, and only bijective functions possess a well-defined inverse function.
3. Composition of Functions and Invertibility
Short Note:
• The composition (gof)(x) = g(f(x)) applies the function f first, followed by g; composition of functions is associative but generally not commutative.
• A function f is invertible if and only if it is bijective, and its inverse f⁻¹ satisfies f⁻¹(f(x)) = x and f(f⁻¹(y)) = y.
NCERT Textbook Exercise Questions
When is a relation R on a set A said to be an equivalence relation?
Answer: A relation R is said to be an equivalence relation if it is simultaneously reflexive (every element is related to itself), symmetric (if a is related to b, then b is related to a), and transitive (if a is related to b and b is related to c, then a is related to c). Equivalence relations divide a set into disjoint equivalence classes.
Show that the function f(x) = 2x is one-one and onto, where f: R → R.
Answer: For one-one: if f(x1) = f(x2), then 2x1 = 2x2, so x1 = x2, proving injectivity. For onto: for any y ∈ R, choosing x = y/2 gives f(x) = y, showing every real number has a preimage. Hence f(x) = 2x is bijective and therefore invertible.
What is the difference between a relation being symmetric and being transitive?
Answer: A relation is symmetric if whenever an element a is related to b, the element b is also related to a. A relation is transitive if whenever a is related to b, and b is related to c, then a must also be related directly to c. These are independent properties; a relation can satisfy one without necessarily satisfying the other.
Why must a function be bijective in order to have an inverse?
Answer: A function must be injective so that each output value corresponds to exactly one input, allowing the inverse to be well-defined without ambiguity. It must also be surjective so that every element of the codomain has a corresponding preimage, ensuring the inverse function is defined for the entire domain of f⁻¹. Only when both conditions hold together can a proper inverse function be constructed.
