Relations and Functions QRT – Quick Revision Tables bring the important concepts, formulas and results of the chapter together in one place. Use these tables for a quick recap while revising, practising questions or preparing for your exams.
Relations and Functions: Chapter 1 Links
Current page: Relations and Functions QRT for Class 12 Maths. Explore other exercises and content from this chapter:
Quick Links to Revision Tables (T): T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | Q&A | PDF
Table 1: Relations and Functions QRT – Basics
| Concept | Formula / Result |
|---|---|
| Relation | A relation R from A to B is an arbitrary subset of the Cartesian product \displaystyle A\times B. |
| Cartesian Product | The Cartesian product \displaystyle A\times B consists of ordered pairs \displaystyle (a,b), where \displaystyle a\in A and \displaystyle b\in B. |
| Relation from A to B | If \displaystyle R\subseteq A\times B, then \displaystyle R is a relation from \displaystyle A to \displaystyle B. |
| Related Elements | If \displaystyle (a,b)\in R, then \displaystyle a is said to be related to \displaystyle b under \displaystyle R, written as \displaystyle aRb. |
| Relation in a Set | A relation in a set A is a subset of \displaystyle A\times A. |
| Empty Relation | A relation \displaystyle R in \displaystyle A is an empty relation if no element of \displaystyle A is related to any element of \displaystyle A, i.e. \displaystyle R=\varnothing\subset A\times A. |
| Universal Relation | A relation \displaystyle R in \displaystyle A is a universal relation if every element of \displaystyle A is related to every element of \displaystyle A, i.e. \displaystyle R=A\times A. |
| Trivial Relations | The empty relation and the universal relation are sometimes called trivial relations. |
Maths Better Tip… Remember: a relation from \displaystyle A to \displaystyle B is simply a subset of \displaystyle A\times B. So every relation is a collection of ordered pairs.
QRT 2: Types of Relations
| Type | Condition / Result |
|---|---|
| Reflexive Relation | A relation R in a set A is reflexive if \displaystyle (a,a)\in R for every \displaystyle a\in A. |
| Symmetric Relation | A relation \displaystyle R in \displaystyle A is symmetric if \displaystyle (a_1,a_2)\in R\Rightarrow(a_2,a_1)\in R for all \displaystyle a_1,a_2\in A. |
| Transitive Relation | A relation \displaystyle R in \displaystyle A is transitive if \displaystyle (a_1,a_2)\in R and \displaystyle (a_2,a_3)\in R\Rightarrow(a_1,a_3)\in R for all \displaystyle a_1,a_2,a_3\in A. |
| Checking Reflexivity | For every \displaystyle a\in A, check whether the ordered pair \displaystyle (a,a) belongs to \displaystyle R. If any such pair is missing, \displaystyle R is not reflexive. |
| Checking Symmetry | Whenever \displaystyle (a,b)\in R, the reverse pair \displaystyle (b,a) must also belong to \displaystyle R. Otherwise, \displaystyle R is not symmetric. |
| Checking Transitivity | Whenever \displaystyle (a,b)\in R and \displaystyle (b,c)\in R, the pair \displaystyle (a,c) must also belong to \displaystyle R. Otherwise, \displaystyle R is not transitive. |
| Three Properties | A relation may possess one, two, all three, or none of the properties: reflexive, symmetric and transitive. |
Maths Better Tip… For quick checking: reflexive means “self”, symmetric means “reverse”, and transitive means “chain”. These three simple ideas make it easier to remember the formal conditions.
Table 3: Relations and Functions QRT – Equivalence Relations
| Concept | Condition / Result |
|---|---|
| Equivalence Relation | A relation \displaystyle R in a set \displaystyle A is an equivalence relation if it is reflexive, symmetric and transitive. |
| Equivalence Relation Test | To show that \displaystyle R is an equivalence relation, verify all three properties: reflexivity, symmetry and transitivity. |
| Equivalence Class | If \displaystyle R is an equivalence relation in \displaystyle A, the equivalence class of \displaystyle a\in A is \displaystyle [a]=\{x\in A:(a,x)\in R\}. |
| Elements of an Equivalence Class | The equivalence class \displaystyle [a] consists of all elements of \displaystyle A that are related to \displaystyle a under \displaystyle R. |
| Class of a Related Element | If \displaystyle aRb, then \displaystyle [a]=[b]. |
| Equivalent Elements | Two elements \displaystyle a,b\in A belong to the same equivalence class precisely when \displaystyle aRb. |
| Distinct Equivalence Classes | Two equivalence classes are either identical or disjoint; they cannot have some common elements and some different elements. |
| Equivalence Classes | The equivalence relation divides the set \displaystyle A into mutually disjoint equivalence classes. |
Maths Better Tip… Remember: an equivalence relation must satisfy all three properties — reflexive, symmetric and transitive. It divides the set into equivalence classes, and any two such classes are either identical or disjoint.
Example: Consider the relation \displaystyle aRb\iff a-b is divisible by \displaystyle 2 on \displaystyle A=\{1,2,3,4\}. Since \displaystyle 1R3, \displaystyle [1]=[3]=\{1,3\}. On the other hand, \displaystyle 1\not R2, and \displaystyle [1]=\{1,3\},\quad [2]=\{2,4\}, so the two equivalence classes are different and disjoint.
Table 4: Relations and Functions QRT – Types of Functions
| Type | Condition / Result |
|---|---|
| One-One (Injective) | A function \displaystyle f:X\to Y is one-one if \displaystyle f(x_1)=f(x_2)\Rightarrow x_1=x_2 for all \displaystyle x_1,x_2\in X. Thus, distinct elements of \displaystyle X have distinct images. |
| Many-One | A function is many-one if two or more distinct elements of the domain can have the same image, i.e. \displaystyle x_1\neq x_2 but \displaystyle f(x_1)=f(x_2) for some \displaystyle x_1,x_2\in X. |
| Onto (Surjective) | A function \displaystyle f:X\to Y is onto if every element of \displaystyle Y is the image of some element of \displaystyle X. Equivalently, \displaystyle \operatorname{Range}(f)=Y. |
| Into | A function \displaystyle f:X\to Y is into if at least one element of the co-domain is not an image of any element of the domain, i.e. \displaystyle \operatorname{Range}(f)\subsetneq Y. |
| Bijective | A function is bijective if it is both one-one and onto. |
| One-One and Onto | For a bijective function, distinct elements of the domain have distinct images and every element of the co-domain is an image of some element of the domain. |
| Finite Set Result | For a finite set \displaystyle X, a function \displaystyle f:X\to X is one-one if and only if it is onto. This result need not hold for infinite sets. |
Note: One-one concerns whether two domain elements can have the same image; onto concerns whether every co-domain element is an image. So always check both conditions separately.
Table 5: Relations and Functions QRT – Composition of Functions
| Concept | Formula / Result |
|---|---|
| Composition of Functions | If \displaystyle f:A\to B and \displaystyle g:B\to C, then the composition of \displaystyle f and \displaystyle g, denoted by \displaystyle g\circ f, is a function from \displaystyle A to \displaystyle C. |
| Composition Formula | \displaystyle (g\circ f)(x)=g(f(x)),\quad \forall x\in A. |
| Order of Composition | In \displaystyle g\circ f, apply \displaystyle f first, followed by \displaystyle g. |
| Domain of \displaystyle g\circ f | The composition \displaystyle g\circ f is defined when the output of \displaystyle f lies in the domain of \displaystyle g. |
| Codomain of \displaystyle g\circ f | If \displaystyle f:A\to B and \displaystyle g:B\to C, then \displaystyle g\circ f:A\to C. |
| Reverse Composition | Similarly, \displaystyle (f\circ g)(x)=f(g(x)), whenever the composition is defined. |
| Composition Is Not Commutative | In general, \displaystyle g\circ f\neq f\circ g, even when both compositions are defined. |
| Identity Function | The identity function on a set \displaystyle A, denoted by \displaystyle I_A, is defined by \displaystyle I_A(x)=x for every \displaystyle x\in A. |
| Composition with Identity | For \displaystyle f:A\to B, \displaystyle f\circ I_A=f and \displaystyle I_B\circ f=f. |
Maths Better Tip… Remember the order: in \displaystyle g\circ f, f acts first and then g. Also, never assume \displaystyle g\circ f=f\circ g; composition of functions is not commutative in general.
View or Download the free Relations and Functions QRT – Class 12 Maths PDF for quick revision anytime.
Table 6: Invertible Functions & Inverse
| Concept | Formula / Result |
|---|---|
| Invertible Function | A function \displaystyle f:A\to B is invertible if there exists a function \displaystyle g:B\to A such that \displaystyle g\circ f=I_A and \displaystyle f\circ g=I_B. |
| Inverse Function | If \displaystyle f:A\to B is invertible, its inverse is denoted by \displaystyle f^{-1}:B\to A and satisfies \displaystyle f^{-1}\circ f=I_A and \displaystyle f\circ f^{-1}=I_B. |
| Condition for Invertibility | A function is invertible if and only if it is both one-one and onto, i.e. bijective. |
| Inverse of an Inverse | If \displaystyle f is invertible, then \displaystyle (f^{-1})^{-1}=f. |
| Inverse of a Composition | If \displaystyle f and \displaystyle g are invertible functions, then \displaystyle (g\circ f)^{-1}=f^{-1}\circ g^{-1}. |
| Inverse of Identity | The identity function is its own inverse: \displaystyle I_A^{-1}=I_A. |
| Inverse Function Values | If \displaystyle f(a)=b, then \displaystyle f^{-1}(b)=a. |
Maths Better Tip… Remember: a function has an inverse only when it is bijective — both one-one and onto. Also, the inverse reverses the direction: \displaystyle f:A\to B gives \displaystyle f^{-1}:B\to A.
Table 7: Relations and Functions QRT – Number of Relations
| Concept | Formula / Result |
|---|---|
| Basic Setup | Let \displaystyle n(A)=m and \displaystyle n(B)=n. |
| Cartesian Product | \displaystyle n(A\times B)=mn. |
| Number of Relations | The number of relations from \displaystyle A to \displaystyle B is \displaystyle 2^{mn}, since every relation is a subset of \displaystyle A\times B. |
| Reflexive Relations | The number of reflexive relations on a set \displaystyle A with \displaystyle m elements is \displaystyle 2^{m^2-m}. |
| Symmetric Relations | The number of symmetric relations on a set \displaystyle A with \displaystyle m elements is \displaystyle 2^{\frac{m(m+1)}{2}}. |
| Transitive Relations | There is no simple general formula for the number of transitive relations on a set with \displaystyle m elements. |
| Equivalence Relations | The number of equivalence relations on a set with \displaystyle m elements is the Bell number \displaystyle B_m. |
| Bell Numbers | \displaystyle B_1=1,\quad B_2=2,\quad B_3=5,\quad B_4=15. |
Note: These counting results apply to finite sets and go beyond the core NCERT syllabus. They are included because they are useful for MCQs, CUET and other competitive-exam questions.
Table 8: Relations and Functions QRT – Number of Functions
| Concept | Formula / Result |
|---|---|
| Basic Setup | Let \displaystyle n(A)=m and \displaystyle n(B)=n. |
| Total Functions | The number of functions from \displaystyle A to \displaystyle B is \displaystyle n^m. |
| One-One Functions | If \displaystyle m\leq n, the number of one-one functions from \displaystyle A to \displaystyle B is \displaystyle {}^nP_m=\frac{n!}{(n-m)!}. |
| One-One: Impossible | If \displaystyle m\gt n, the number of one-one functions is \displaystyle 0, since the domain has more elements than the co-domain. |
| Many-One Functions | Number of many-one functions = \displaystyle \text{Total functions}-\text{one-one functions}. |
| Onto: Equal Size | If \displaystyle m=n, the number of onto functions is \displaystyle n!. |
| Onto: Impossible | If \displaystyle m\lt n, the number of onto functions is \displaystyle 0, since the domain has fewer elements than the co-domain. |
| Onto: General Case | If \displaystyle m\gt n, the number of onto functions is \displaystyle \sum_{k=0}^{n}(-1)^k\,{}^{n}C_k\,(n-k)^m, by the Inclusion–Exclusion Principle. |
| Into Functions | Number of into functions = \displaystyle \text{Total functions}-\text{onto functions}. |
| Bijective Functions | Bijective functions are possible only when \displaystyle m=n; their number is \displaystyle n!. |
Maths Better Tip… For counting functions, remember the basic order: total → one-one → onto → into → bijective. First compare the sizes of the domain and co-domain to see which types are possible.
Table 9: Relations and Functions QRT – Important Results & Quick Recall
| Concept | Result / Quick Recall |
|---|---|
| Intersection of Equivalence Relations | If \displaystyle R_1 and \displaystyle R_2 are equivalence relations in a set \displaystyle A, then \displaystyle R_1\cap R_2 is also an equivalence relation. |
| Relation Induced by a Function | For a function \displaystyle f:X\to Y, the relation defined by \displaystyle aRb\iff f(a)=f(b) is an equivalence relation on \displaystyle X. |
| One-One Function on a Finite Set | A one-one function from a finite set \displaystyle A to itself is a permutation of the elements of \displaystyle A. |
| Equality of Functions | Two functions \displaystyle f:A\to B and \displaystyle g:A\to B are equal if \displaystyle f(a)=g(a) for every \displaystyle a\in A. |
| Sum of One-One Functions | The sum of two one-one functions need not be one-one. |
| Sum of Onto Functions | The sum of two onto functions need not be onto. |
| Composition and Invertibility | If \displaystyle f:A\to B has an inverse \displaystyle g:B\to A, then \displaystyle g\circ f=I_A and \displaystyle f\circ g=I_B. |
| Invertibility Test | A function is invertible if and only if it is both one-one and onto. |
Maths Better Tip… Do not assume that familiar properties are preserved under function operations. In particular, the sum of two one-one functions need not be one-one, and the sum of two onto functions need not be onto.
Quick Questions — Test Yourself
Maths Better Tip… Try to answer each question before opening it. Then click the arrow to reveal the answer and check yourself.
Continue Your Revision
Want to practise further? Explore the detailed NCERT Solutions for Relations and Functions & other chapters and also, continue learning with the Maths Better YouTube playlist on Relations and Functions. Use the Relations and Functions QRTs for a quick recap, and return to the detailed solutions whenever you need step-by-step practice.
All the best and keep learning 👍
📘 Relations and Functions QRT – Class 12 Maths
9 Quick Revision Tables + Quick Q&A
