Sets, Relations & Functions: JEE Main Complete Guide

Sets, Relations and Functions is the foundational chapter of JEE Main Mathematics, but it is tested in two to three questions every session at a surprisingly sophisticated level — not with the basic Venn diagram questions of school, but with function classification, domain and range computation, composition, and the analysis of special function types (periodic, odd/even, invertible). This guide covers all the concepts JEE actually tests, going beyond the basics that most revision guides stop at.

Sets: Counting and Venn Diagram Formulae

For three sets A, B, C: |A∪B∪C| = |A| + |B| + |C| − |A∩B| − |B∩C| − |A∩C| + |A∩B∩C|. JEE uses this in word problems about survey responses or intersecting groups — draw a Venn diagram and fill in the regions from the inside out. The number of subsets of a set with n elements is 2ⁿ; proper subsets is 2ⁿ − 1; non-empty subsets is 2ⁿ − 1; power set has 2ⁿ elements. Relations: a relation R on A is reflexive if (a,a) ∈ R ∀a ∈ A; symmetric if (a,b) ∈ R → (b,a) ∈ R; transitive if (a,b) and (b,c) ∈ R → (a,c) ∈ R; an equivalence relation is all three.

The number of equivalence relations on a set of n elements equals the Bell number B_n. For small n: B₁=1, B₂=2, B₃=5, B₄=15 — these exact values appear in JEE. Take a free sets and functions mock to test your counting speed.

Domain and Range Computation

Finding the domain of f(x) = g(x)/h(x): set h(x) ≠ 0 and intersect with the domain of g(x). For f(x) = √g(x): require g(x) ≥ 0. For f(x) = log g(x): require g(x) > 0. For composite f(g(x)): first find the range of g(x) in the domain of g, then restrict to inputs that the outer function f can accept. This two-step domain-of-composite procedure is the most tested domain technique in JEE.

Range computation: for a rational function, let y = f(x) and solve for x — the range is all y for which the equation in x has a real solution. This "solve for x" method transforms a range question into a discriminant condition. For f(x) = (ax+b)/(cx+d), the range is all y except a/c (the horizontal asymptote). For quadratic functions, the range uses the vertex formula: if f(x) = ax² + bx + c, range is [f(−b/2a), ∞) for a > 0 or (−∞, f(−b/2a)] for a < 0.

Special Function Types and Composition

Odd function: f(−x) = −f(x), graph symmetric about origin. Even function: f(−x) = f(x), graph symmetric about y-axis. Neither is possible (e.g., general linear functions). Products of same-parity functions are even (odd × odd = even); products of different parity are odd. Definite integral of odd function on symmetric interval = 0 — this connects directly to the integration techniques in our definite integration guide.

Periodic functions: f(x+T) = f(x) for all x in the domain, where T is the period. The period of sinx, cosx = 2π; tanx, cotx = π; |sinx|, |cosx| = π. For composite periods: period of f(g(x)) requires careful analysis. Period of sin(nx) = 2π/n; period of sin(πx) = 2; period of f(x)+g(x) = LCM(period of f, period of g) when both are periodic. The LCM rule for combined periods appears as a direct JEE question. Invertible functions: f is invertible if and only if it is bijective (one-to-one and onto). The inverse function swaps domain and range; f⁻¹(f(x)) = x. Composition: (f∘g)(x) = f(g(x)); in general f∘g ≠ g∘f. The domain of f∘g is {x in domain of g : g(x) is in domain of f}. For the complete algebra and function framework, see our Math 2026 strategy guide and our matrices guide.