Math · Sets, Relations and Functions

Relations and Functions formulas for JEE

Every Relations and Functions formula you need for JEE, grouped by concept.

FormulasRevision notes
18 formulas4 concepts
01Relations02Functions and their Representation03Operations on Functions04Types of Real Functions
01

Relations

9 formulas

Mathematical Relation

R⊆A×BR \subseteq A \times B

A relation R from set A to set B is a subset of the Cartesian product.

applies whenA and B are non-empty sets.
relationscartesian_product

Empty Relation

R=∅⊆A×AR = \emptyset \subseteq A \times A

A relation where no element of A is related to any element of A.

applies whenDefined on set A.
relationsempty

Reflexive Relation

(a,a)∈R,∀a∈A(a, a) \in R, \forall a \in A

A relation in which every element is related to itself.

applies whenMust hold for every element in set A.
relationsreflexive

Symmetric Relation

(a,b)∈R  ⟹  (b,a)∈R(a, b) \in R \implies (b, a) \in R

A relation where the order of related pairs is reversible.

applies whenFor all a, b in A.
relationssymmetric

Total Number of Relations

N=2mnN = 2^{mn}

The total possible number of relations from set A to set B.

applies when∣A∣=m,∣B∣=n|A| = m, |B| = n
relationscombinatoricsjee-advanced

Total Number of Reflexive Relations

N=2n(n−1)N = 2^{n(n-1)}

The total possible number of reflexive relations on a set A.

applies when∣A∣=n|A| = n
relationscombinatoricsreflexivejee-advanced

Total Number of Symmetric Relations

N=2n(n+1)2N = 2^{\frac{n(n+1)}{2}}

The total possible number of symmetric relations on a set A.

applies when∣A∣=n|A| = n
relationscombinatoricssymmetricjee-advanced

Transitive Relation

(a,b)∈R∧(b,c)∈R  ⟹  (a,c)∈R(a, b) \in R \land (b, c) \in R \implies (a, c) \in R

A relation where a chain of relations implies a direct relation.

applies whenFor all a, b, c in A. Vacuously true if (b,c) does not exist.
relationstransitive

Universal Relation

R=A×AR = A \times A

A relation where every element of A is related to every element of A.

applies whenDefined on set A.
relationsuniversal
02

Functions and their Representation

4 formulas

One-One (Injective) Function

f(x1)=f(x2)  ⟹  x1=x2f(x_1) = f(x_2) \implies x_1 = x_2

A function where distinct elements have distinct images.

applies whenFor all x1, x2 in the domain X.
functionsinjective

Onto (Surjective) Function Definition

∀y∈Y,∃x∈X s.t. f(x)=y\forall y \in Y, \exists x \in X \text{ s.t. } f(x) = y

A function where every element in the co-domain is mapped to by at least one element in the domain.

applies whenDomain X, Co-domain Y.
functionssurjective

Onto Function (Range Condition)

Range of f=Y\text{Range of } f = Y

A function is onto if and only if its range equals its co-domain.

applies whenCo-domain is Y.
functionssurjectiverange

Total Number of Bijective Functions

N=n!N = n!

The total possible number of bijective functions from a set A to itself.

applies when∣A∣=∣B∣=n|A| = |B| = n
functionscombinatoricsbijective
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03

Operations on Functions

2 formulas

Composition of Functions

(g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))

The application of one function to the result of another.

applies whenRange of f must be a subset of the Domain of g.
functionscomposition

Invertible Function Condition

g∘f=IX∧f∘g=IY  ⟹  g=f−1g \circ f = I_X \land f \circ g = I_Y \implies g = f^{-1}

A function g is the inverse of f if their composition yields the identity function.

applies whenf must be bijective (one-one and onto).
functionsinverse
04

Types of Real Functions

3 formulas

Greatest Integer Function

f(x)=[x]f(x) = [x]

Returns the greatest integer less than or equal to x.

applies whenx∈Rx \in \mathbb{R}
functionsgreatest_integerstep_function

Modulus Function

∣x∣={x,x≥0−x,x<0|x| = \begin{cases} x, & x \ge 0 \\\\ -x, & x < 0 \end{cases}

Returns the non-negative magnitude of a real number.

applies whenx∈Rx \in \mathbb{R}
functionsmodulusabsolute_value

Signum Function

sgn(x)={1,x>00,x=0−1,x<0\text{sgn}(x) = \begin{cases} 1, & x > 0 \\\\ 0, & x = 0 \\\\ -1, & x < 0 \end{cases}

Extracts the sign of a real number.

applies whenx∈Rx \in \mathbb{R}
functionssignum
Other chapters
MathSets34 formulasMathTrigonometric Functions56 formulasMathComplex Numbers and Quadratic Equations36 formulasMathPermutations and Combinations21 formulasMathBinomial Theorem24 formulasMathSequence and Series25 formulasMathStraight Lines27 formulasMathConic Sections35 formulasMathIntroduction to Three-dimensional Geometry9 formulasMathLimits and Derivatives32 formulasMathStatistics25 formulasMathProbability17 formulasMathRelations and Functions14 formulasMathInverse Trigonometric Functions33 formulasMathMatrices44 formulasMathDeterminants23 formulasMathContinuity and Differentiability30 formulasMathApplications of Derivatives26 formulasMathIntegrals47 formulasMathApplications of the Integrals14 formulasMathDifferential Equations23 formulasMathVectors29 formulasMathThree-dimensional Geometry32 formulasMathProbability15 formulas

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