Sets, Relations and Functions – JEE Main Maths Formula Sheet & Class 11 Notes | All Formulas, Laws & Results

Sets — Definition, Notation, Types, and Subsets Formulas

Why Set Types and Subset Formulas Are Direct JEE Main Maths Questions

A set in Sets, Relations and Functions is a well-defined collection of distinct objects. Sets are denoted by capital letters (A, B, C) and elements by lowercase letters. The symbol ∈ denotes "belongs to" and ∉ denotes "does not belong to." Sets can be written in Roster form (elements listed in braces, e.g., {1, 2, 3}) or Set-builder form (e.g., {x : x is a natural number less than 4}).

Standard number sets (Sets, Relations and Functions notation): ℕ = {1, 2, 3, …} (natural numbers); W = {0, 1, 2, 3, …} (whole numbers); ℤ or I = {…, –2, –1, 0, 1, 2, …} (integers); ℚ = {p/q : p, q ∈ ℤ, q ≠ 0} (rationals); ℝ = all real numbers; ℂ = {a + ib : a, b ∈ ℝ, i = √–1} (complex numbers). Order: ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ. Irrational numbers: non-repeating, non-terminating decimals (√2, √3, π, e).

Types of sets in Sets, Relations and Functions: Null/Empty set — denoted ∅ or {}; has no elements; n(∅) = 0; ∅ is a subset of every set. Singleton set — exactly one element. Finite set — countable elements; n(A) = finite number. Infinite set — ℕ, ℤ, ℚ, ℝ all infinite. Equal sets — A = B iff A ⊆ B and B ⊆ A (same elements, order immaterial). Equivalent sets — n(A) = n(B) (same number of elements but not necessarily same elements). Equal sets are always equivalent; equivalent sets may not be equal.

Subset formulas in Sets, Relations and Functions: A ⊆ B means every element of A is in B. A ⊂ B (proper subset) means A ⊆ B and B has at least one element not in A. Every set is a subset of itself (A ⊆ A) but not a proper subset (A ⊄ A). ∅ ⊆ A for every set A. If n(A) = n, then: number of subsets of A = 2ⁿ; number of proper subsets = 2ⁿ – 1. Power set P(A) = set of all subsets of A; n(P(A)) = 2ⁿ. If A ⊆ B then P(A) ⊆ P(B). If A = ∅ then P(A) = {∅} — singleton set. P(A) ∩ P(B) = P(A ∩ B); P(A) ∪ P(B) ⊆ P(A ∪ B).

Intervals as subsets of ℝ (Sets, Relations and Functions formulas): Open interval (a, b) = {x : a < x < b}. Closed interval [a, b] = {x : a ≤ x ≤ b}. Half-open (a, b] = {x : a < x ≤ b}. Half-closed [a, b) = {x : a ≤ x < b}. Infinite intervals: (a, ∞) = {x : x > a}; (–∞, b) = {x : x < b}; (–∞, ∞) = ℝ; (0, ∞) = ℝ⁺; (–∞, 0) = ℝ⁻. Download the Free PDF for Sets, Relations and Functions for the complete types of sets summary.

Set Operations, De Morgan's Laws, and Algebra of Sets Formulas

Why De Morgan's Laws and Set Operation Formulas Are Core JEE Main Maths Topics

The fundamental operations on sets in Sets, Relations and Functions define how sets combine. These operations have properties — the Algebra of Sets — that are tested directly in JEE Main maths through symbolic manipulation and Venn diagram questions.

Union A ∪ B = {x : x ∈ A or x ∈ B}. Intersection A ∩ B = {x : x ∈ A and x ∈ B}. Difference A – B = {x : x ∈ A and x ∉ B} (also written A ∼ B or A/B). Symmetric Difference A △ B = (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B). Complement A' = U – A = {x : x ∈ U, x ∉ A}. Key results: A ∩ A' = ∅; A ∪ A' = U; (A')' = A; ∅' = U; U' = ∅.

Algebra of Sets laws (Sets, Relations and Functions formulas): Idempotent: A ∪ A = A; A ∩ A = A. Identity: A ∪ ∅ = A; A ∩ ∅ = ∅; A ∪ U = U; A ∩ U = A. Commutative: A ∪ B = B ∪ A; A ∩ B = B ∩ A. Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C); (A ∩ B) ∩ C = A ∩ (B ∩ C). Distributive: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

De Morgan's Laws (Sets, Relations and Functions): (A ∪ B)' = A' ∩ B'. (A ∩ B)' = A' ∪ B'. Extended: A – (B ∪ C) = (A – B) ∩ (A – C). A – (B ∩ C) = (A – B) ∪ (A – C). These De Morgan's Laws in Sets, Relations and Functions are among the most-tested results in JEE Main maths and CBSE Class 11 boards. Important set identities for JEE Main: A – B = A ∩ B'; (A – B) ∪ B = A ∪ B; (A – B) ∩ B = ∅; A △ B = B △ A (symmetric difference is commutative); A △ ∅ = A; A △ A = ∅; A △ B = ∅ ⟺ A = B; A ∩ (B △ C) = (A ∩ B) △ (A ∩ C). Download the Free PDF for the complete algebra of sets formula list for Sets, Relations and Functions JEE Main.

Cardinal Number Formulas — Two and Three Sets (JEE Main Favourite)

Why Cardinal Number Formulas Are the Most-Tested Sets, Relations and Functions Results in JEE Main

Cardinal number formulas in Sets, Relations and Functions give the number of elements in set combinations. These formulas appear in almost every JEE Main maths session either as direct calculation questions or as the basis for problem-solving in word problems involving surveys, groups, or logical deductions.

Cardinal number formulas for two sets (Sets, Relations and Functions):

n(A ∪ B) = n(A) + n(B) – n(A ∩ B) — the most fundamental formula in Sets, Relations and Functions.

n(A ∩ B) = n(A) + n(B) – n(A ∪ B)

n(A – B) = n(A) – n(A ∩ B) = n(A ∪ B) – n(B)

n(A △ B) = n(A) + n(B) – 2·n(A ∩ B) = n(A ∪ B) – n(A ∩ B)

n(A' ∩ B') = n(A ∪ B)' = n(U) – n(A ∪ B)

n(A' ∪ B') = n(A ∩ B)' = n(U) – n(A ∩ B)

If A and B are disjoint: n(A ∪ B) = n(A) + n(B)

Cardinal number formulas for three sets (Sets, Relations and Functions — most important for JEE Main):

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C) + n(A∩B∩C)

Number of elements in exactly one of A, B, C = n(A) + n(B) + n(C) – 2n(A∩B) – 2n(B∩C) – 2n(A∩C) + 3n(A∩B∩C)

Number of elements in exactly two of A, B, C = n(A∩B) + n(B∩C) + n(A∩C) – 3·n(A∩B∩C)

Number of elements in at least two of A, B, C = n(A∩B) + n(B∩C) + n(A∩C) – 2·n(A∩B∩C)

Number of elements in at most two of A, B, C = n(A∪B∪C) – n(A∩B∩C)

n(A ∩ B') = n(A) – n(A ∩ B) [elements only in A, not B]

Key Cartesian product cardinal results: n(A × B) = n(A) · n(B). n(P(A × B)) = 2^(n(A)·n(B)). If n(A ∩ B) = k then n(A×B ∩ B×A) = k². A × (B ∪ C) = (A×B) ∪ (A×C). A × (B ∩ C) = (A×B) ∩ (A×C). A × (B – C) = (A×B) – (A×C). (A×B) ∩ (C×D) = (A∩C) × (B∩D). Download the Free PDF for Sets, Relations and Functions for all cardinal number formulas with worked examples.

Relations — Types, Properties, and Equivalence Relation Formulas

Why Relation Types Are a JEE Main Classification Question in Sets, Relations and Functions

A relation R from set A to set B is any subset of A × B. Relations are the bridge between sets and functions in Sets, Relations and Functions — understanding relation properties is essential for identifying function types in JEE Main maths.

Domain, Range, Co-domain of a Relation in Sets, Relations and Functions: Domain of R = set of first elements of all ordered pairs in R. Range of R = set of second elements of all ordered pairs in R. Co-domain = set B (for R ⊆ A×B). Range ⊆ Co-domain always. For R = A×B: Dom(R) = A, Range(R) = B. For the inverse relation R⁻¹: Domain of R⁻¹ = Range of R; Range of R⁻¹ = Domain of R; (R⁻¹)⁻¹ = R. If n(A) = p and n(B) = q then n(A×B) = pq and total number of possible relations from A to B = 2^(pq).

Types of relations on a set (Sets, Relations and Functions — R ⊆ A×A):

Reflexive relation: (a, a) ∈ R for all a ∈ A. Every element is related to itself. Identity relation ⊆ reflexive relation. Example: "is equal to", "is parallel to" — reflexive.

Symmetric relation: (a, b) ∈ R ⟹ (b, a) ∈ R for all a, b ∈ A. Example: "is parallel to", "is a sibling of."

Transitive relation: (a, b) ∈ R and (b, c) ∈ R ⟹ (a, c) ∈ R. Example: "is parallel to", "is a subset of", "is less than."

Anti-symmetric relation: (a, b) ∈ R and (b, a) ∈ R ⟹ a = b. Example: "is a subset of (⊆)", "is less than or equal to (≤)".

Equivalence relation in Sets, Relations and Functions: A relation R on A is an equivalence relation if and only if it is simultaneously Reflexive, Symmetric, AND Transitive. Examples: "is equal to", "is parallel to", "is congruent to", "is similar to". Every identity relation (I_A = {(a,a): a ∈ A}) is an equivalence relation but not vice versa. Void relation (∅) is symmetric and transitive but not reflexive (unless A = ∅). Universal relation (A×A) is always reflexive, symmetric, and transitive — hence an equivalence relation. Number of equivalence relations on a set of n elements = Bell number B(n): B(1)=1, B(2)=2, B(3)=5, B(4)=15. Download the Free PDF for Sets, Relations and Functions for the complete relation type classification table.

Functions — Definition, Types, and Counting Formulas in Sets, Relations and Functions

Why Function Classification Is the Most-Tested Sets, Relations and Functions Topic in JEE Main Maths

A function f : A → B in Sets, Relations and Functions is a special relation where every element of A (domain) has exactly one image in B (co-domain). The set of all actual images is the range. Range ⊆ Co-domain always. A relation is a function if no vertical line intersects its graph more than once (vertical line test).

One-One (Injective) function in Sets, Relations and Functions: Different elements in A have different images in B: f(x₁) = f(x₂) ⟹ x₁ = x₂. For a differentiable function: if f'(x) > 0 for all x, or f'(x) < 0 for all x (strictly monotone) → one-one. Linear functions (ax+b, a≠0), exponential (eˣ), logarithmic (log x), identity (x) are always one-one. Even functions, periodic functions, modulus are many-one. Number of one-one functions from A to B: if n(A) = m, n(B) = n, then = ⁿPₘ if n ≥ m; = 0 if n < m.

Onto (Surjective) function in Sets, Relations and Functions: Every element of co-domain B has at least one pre-image in A. Range = Co-domain. Number of onto functions from A (m elements) to B (n elements) = Σ(–1)^(n–k) · C(n,k) · k^m (inclusion-exclusion). If m < n → no onto function possible.

Bijective function in Sets, Relations and Functions: Both one-one AND onto. Number of bijective functions from A to B (when n(A) = n(B) = n) = n! (n factorial). Only bijective functions have inverse functions. For bijective f: A → B, the inverse f⁻¹: B → A satisfies f(f⁻¹(y)) = y and f⁻¹(f(x)) = x. Graph of f⁻¹ is the reflection of graph of f about the line y = x.

Many-one function: two or more elements of A have the same image in B. All even functions, all periodic functions, trigonometric functions (in their full natural domain), square function are many-one.

Into function: at least one element in co-domain has no pre-image. Range ⊊ Co-domain (strictly smaller). Constant functions are many-one into functions.

Composite function gof in Sets, Relations and Functions: gof(x) = g(f(x)). gof is defined only if Range(f) ⊆ Domain(g). In general gof ≠ fog (composition is not commutative). If f is one-one and g is one-one → gof is one-one. If f is onto and g is onto → gof is onto. If gof is one-one → f is one-one. If gof is onto → g is onto. Download the Free PDF for Sets, Relations and Functions for the complete function type counting table.

Domain and Range of Standard Functions — Complete Table for JEE Main

Why Domain and Range of Functions Is the Most Direct Sets, Relations and Functions JEE Main Question

Finding the domain and range of functions in Sets, Relations and Functions is one of the most tested skills in JEE Main maths. The domain is the set of all x for which f(x) exists (is real, finite, and defined). The range is the set of all actual output values.

Standard function domain and range table (Sets, Relations and Functions JEE Main):

x^(2n+1), n∈W (odd power): Domain = ℝ; Range = ℝ.

x^(2n), n∈ℕ (even power): Domain = ℝ; Range = [0, ∞).

√x: Domain = [0, ∞); Range = [0, ∞).

1/x: Domain = ℝ–{0}; Range = ℝ–{0}.

1/√x: Domain = (0, ∞); Range = (0, ∞).

aˣ (a > 0, a ≠ 1): Domain = ℝ; Range = (0, ∞).

logₐ x (a > 0, a ≠ 1): Domain = (0, ∞) = ℝ⁺; Range = ℝ.

sin x: Domain = ℝ; Range = [–1, 1].

cos x: Domain = ℝ; Range = [–1, 1].

tan x: Domain = ℝ – {(2n+1)π/2, n∈ℤ}; Range = ℝ.

cot x: Domain = ℝ – {nπ, n∈ℤ}; Range = ℝ.

sec x: Domain = ℝ – {(2n+1)π/2, n∈ℤ}; Range = (–∞, –1] ∪ [1, ∞).

cosec x: Domain = ℝ – {nπ, n∈ℤ}; Range = (–∞, –1] ∪ [1, ∞).

sin⁻¹x: Domain = [–1, 1]; Range = [–π/2, π/2].

cos⁻¹x: Domain = [–1, 1]; Range = [0, π].

tan⁻¹x: Domain = ℝ; Range = (–π/2, π/2).

cot⁻¹x: Domain = ℝ; Range = (0, π).

sec⁻¹x: Domain = (–∞,–1]∪[1,∞); Range = [0,π] – {π/2}.

cosec⁻¹x: Domain = (–∞,–1]∪[1,∞); Range = [–π/2, π/2] – {0}.

|x| (modulus): Domain = ℝ; Range = [0, ∞).

[x] (greatest integer): Domain = ℝ; Range = ℤ.

{x} (fractional part): Domain = ℝ; Range = [0, 1).

Signum sgn(x): Domain = ℝ; Range = {–1, 0, 1}.

Rules for finding domain (Sets, Relations and Functions JEE Main): For √f(x): require f(x) ≥ 0. For 1/f(x): require f(x) ≠ 0. For log f(x): require f(x) > 0. For √(f(x))/g(x): require f(x) ≥ 0 AND g(x) ≠ 0. For asin(f(x)) or acos(f(x)): require –1 ≤ f(x) ≤ 1. For tan f(x): require f(x) ≠ π/2 + nπ. Range finding methods: completing the square, using AM ≥ GM, discriminant condition (D ≥ 0 for real solution), and for a sin θ + b cos θ: range = [–√(a²+b²), √(a²+b²)]. Download the Free PDF for the full domain-range table of all standard functions in Sets, Relations and Functions JEE Main.

Special Functions — Modulus, Greatest Integer, Fractional Part, Signum, Even/Odd, Periodic

Why Special Function Properties Are High-Frequency JEE Main Maths Questions on Sets, Relations and Functions

Special functions in Sets, Relations and Functions have properties and formulas that appear regularly in JEE Main maths as direct application questions. Mastering these properties of special functions is essential for the Sets, Relations and Functions formula sheet.

Modulus function |x| (Sets, Relations and Functions): f(x) = |x| = x if x ≥ 0; = –x if x < 0. Domain = ℝ; Range = [0, ∞). Properties: |xy| = |x||y|; |x/y| = |x|/|y| (y ≠ 0); |x+y| ≤ |x|+|y| (triangle inequality); ||x|–|y|| ≤ |x+y|. Solution rules: |f(x)| = a (a > 0) ⟹ f(x) = ±a. |f(x)| ≤ a ⟹ –a ≤ f(x) ≤ a. |f(x)| ≥ a ⟹ f(x) ≤ –a or f(x) ≥ a. Minimum of |x–a| + |x–b| = |a–b|, attained for a ≤ x ≤ b. |x+y| = |x|+|y| iff xy ≥ 0. |x–y| = |x|+|y| iff xy ≤ 0.

Greatest Integer function [x] (floor function) in Sets, Relations and Functions: [x] = n if n ≤ x < n+1, n ∈ ℤ. Domain = ℝ; Range = ℤ. Examples: [3.7]=3, [–3.7]=–4, [3]=3. Properties of [x]: [x+n] = [x]+n (n ∈ ℤ); [x] + [–x] = 0 if x ∈ ℤ; = –1 if x ∉ ℤ; x–1 < [x] ≤ x; [x] ≥ n ⟺ x ≥ n; [x] < n ⟺ x < n; [x] > n ⟺ x ≥ n+1. [x+y] ≥ [x]+[y]; [x/n] = [[x]/n].

Fractional Part function {x} in Sets, Relations and Functions: {x} = x – [x]. Domain = ℝ; Range = [0, 1). Fundamental period = 1. {x} = 0 if x ∈ ℤ; 0 < {x} < 1 if x ∉ ℤ.

Signum function (Sets, Relations and Functions): sgn(x) = 1 if x > 0; = 0 if x = 0; = –1 if x < 0. Also written as |x|/x for x ≠ 0. Domain = ℝ; Range = {–1, 0, 1}.

Even and Odd functions (Sets, Relations and Functions): Even: f(–x) = f(x) — symmetric about y-axis. Odd: f(–x) = –f(x) — symmetric about origin. f(x) = 0 is the only function that is both even and odd. Any function can be written as sum of even and odd parts: f(x) = [f(x)+f(–x)]/2 + [f(x)–f(–x)]/2. Even+Even=Even; Odd+Odd=Odd; Even×Even=Even; Odd×Odd=Even; Even×Odd=Odd; Even+Odd = Neither. Derivative of even function is odd; derivative of odd is even.

Periodic functions (Sets, Relations and Functions): f(x+T) = f(x) for all x in domain, T > 0 — T is a period. Smallest positive T = fundamental period. Period of sin x, cos x, cosec x, sec x = 2π. Period of tan x, cot x = π. Period of |sin x|, |cos x| = π. Period of {x} = 1. Period of [x] — not periodic. Constant function — periodic but no fundamental period. If f has period T: f(ax+b) has period T/|a|. If f has period T₁ and g has period T₂: period of f+g = LCM(T₁, T₂) or a divisor of it. LCM of fractions: LCM(p/q, r/s) = LCM(p,r)/HCF(q,s). LCM of rational and irrational is not defined in general. Download the Free PDF for all Sets, Relations and Functions special function properties and period formulas.

Logarithmic and Exponential Functions — Key Properties for Sets, Relations and Functions JEE Main

Why Log and Exponential Function Properties Are Formula-Intensive JEE Main Topics

Logarithmic and exponential functions appear throughout JEE Main maths — in Sets, Relations and Functions for domain problems, and in limits, differentiation, and integration later. Mastering their properties from the Sets, Relations and Functions formula sheet provides a foundation for the entire JEE Main maths syllabus.

Properties of logarithmic functions (Sets, Relations and Functions): logₐ(mn) = logₐm + logₐn. logₐ(m/n) = logₐm – logₐn. logₐ(mⁿ) = n·logₐm. logₐb = (log b)/(log a) = logₓb / logₓa (change of base). logₐb · logᵦa = 1. logₐb = 1/logᵦa. (logₐb)^p / q = logₐ(b^(p/q)). logₐa = 1. logₐ1 = 0. Domain of logₐx: x > 0, a > 0, a ≠ 1. If a > 1: logₐf(x) ≥ logₐg(x) ⟺ f(x) ≥ g(x) > 0 (increasing). If 0 < a < 1: logₐf(x) ≥ logₐg(x) ⟺ 0 < f(x) ≤ g(x) (decreasing).

Algebraic properties of functions (Sets, Relations and Functions — identification results for JEE Main): If f(x+y) = f(x)+f(y) ⟹ f(x) = kx (linear). If f(x+y) = f(x)·f(y) ⟹ f(x) = aˣ (exponential). If f(xy) = f(x)+f(y) (x,y>0) ⟹ f(x) = logₐx (logarithmic). If f(x/y) = f(x)–f(y) (x,y>0) ⟹ f(x) = logₐx. If f(x)·f(1/x) = f(x)+f(1/x) ⟹ f(x) = xⁿ±1 or f(x) = 1±xⁿ. If f(x)±f(y) = f(1/(xy∓1)) ⟹ f(x) = tan⁻¹x. If f'(x) = f(x) ⟹ f(x) = keˣ. These algebraic property identifiers appear as direct one-mark questions in JEE Main maths every year from the Sets, Relations and Functions section. Download the Free PDF for all Sets, Relations and Functions logarithm formulas and function identifiers.