Class 12 Maths · Chapter 1
Unit: Relations and Functions
Tentative Study Share: 4.0 Marks
Chapter 1: Relations And Functions — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Relations And Functions.
NCERT Exercises in Chapter 1: Relations And Functions
📘 NCERT Concept Notes — Relations And Functions
01 · A relation is a set of ordered pairs
Start by naming the carrier set. Reflexivity checks every self-pair, symmetry checks reverses, and transitivity checks every two-pair chain. To disprove a property, one valid counterexample is enough. A chain that never occurs cannot disprove transitivity.
02 · Equivalence groups elements
An equivalence relation behaves like equality within groups. Its classes collect all elements related to a chosen element. Check all three properties before describing classes.
03 · A function needs its domain and codomain
Injectivity asks whether distinct inputs can share an output. Surjectivity asks whether every codomain element is reached. The same expression can change classification when the declared sets change.
04 · Composition follows the arrows
For g∘f, apply f first and g second. An inverse must undo the function in both directions, on the declared sets. It is not the reciprocal of the function value.
⚡ Key Formulas & Results — Relations And Functions
Three relation tests
What must you test for reflexive, symmetric and transitive relations?
\(\begin{aligned}&aRa\quad(\forall a\in A)\\&aRb\Rightarrow bRa\\&(aRb\land bRc)\Rightarrow aRc\end{aligned}\)
Empty and universal relations
What are the smallest and largest possible relations on A?
\(R_{empty}=\varnothing,\quad R_{universal}=A\times A\)
Equivalence and classes
Which properties make an equivalence relation?
\(R\text{ reflexive, symmetric, transitive};\quad[a]=\{b\in A:aRb\}\)
Derive the parity classes
Why does even difference define an equivalence relation?
\([1]=\{1,3,5\},\quad[2]=\{2,4\}\)
One-one
What implication proves injectivity?
\(f(x_1)=f(x_2)\Rightarrow x_1=x_2\)
Onto and bijective
What must an onto proof produce for an arbitrary output?
\(\forall y\in B\ \exists x\in A:f(x)=y\)
Order of composition
Which function acts first in g∘f?
\((g\circ f)(x)=g(f(x))\)
Inverse function
What two identities certify an inverse?
\(f^{-1}\circ f=I_A,\quad f\circ f^{-1}=I_B\)
Derive the textbook inverse
How does solving y=4x+3 give the inverse?
\(f^{-1}(y)=\frac{y-3}{4}\)
🎯 CBSE 2026 Board PYQs from Relations And Functions
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q32
A relation R is defined on Z, the set of integers, as R = {(x, y) : |x − y| is divisible by a prime…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q34
A relation R is defined on Z, the set of integers, as R = {(x, y) : |x − y| is divisible by a prime…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q33
A relation R is defined on Z, the set of integers, as R = {(x, y) : |x − y| is divisible by a prime…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q1
On A={1,2,3}, classify R={(1,1),(2,2),(1,2)}.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q21
Is f: ℝ∖{3}→ℝ, f(x)=(x−2)/(x−3), onto? OR Is f: ℤ×ℤ→ℤ×ℤ, f(x,y)=(2y,3x), injective?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q12
On A={1,2,3}, classify R={(1,2),(2,1),(2,2)}.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q24
Is f: ℝ∖{3}→ℝ, f(x)=(x−2)/(x−3), onto? OR Is f: ℤ×ℤ→ℤ×ℤ, f(x,y)=(2y,3x), injective?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q5
On A={1,2,3}, classify R={(1,3),(3,3),(1,1),(2,2),(3,1)}.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q22
Is f: ℝ∖{3}→ℝ, f(x)=(x−2)/(x−3), onto? OR Is f: ℤ×ℤ→ℤ×ℤ, f(x,y)=(2y,3x), injective?
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q32
Show that \(f:\mathbb R\to\mathbb R\), \(f(x)=x/\sqrt{1+x^2}\), is one-one but not onto.
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q34
Show that f:[0,∞)→[−5,∞), f(x)=4x²+4x−5 is one-one and onto. Find p≥0 with f(p)=3.
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q33
The printed problem specifies f:[0,∞)→A⊂ℕ, f(x)=4x²+12x+15. Examine whether it defines a function with…
CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Q20
Assertion: f:ℕ→ℕ, f(x)=x³+2 is injective but not onto. Reason: For every natural y there is no natural…
CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Q26
Is f:ℝ∖{3}→ℝ∖{1}, f(x)=(x−2)/(x−3), one-one and onto? OR For a fixed positive integer n, determine…
CBSE Class 12 Mathematics 2026 Main Paper 65/4/2 · Q20
Assertion: f:ℕ→ℕ, f(x)=x³+2 is injective but not onto. Reason: For every natural y there is no natural…
CBSE Class 12 Mathematics 2026 Main Paper 65/4/2 · Q28
Is f:ℝ∖{3}→ℝ∖{1}, f(x)=(x−2)/(x−3), one-one and onto? OR For a fixed positive integer n, determine…
CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Q20
Assertion: f:ℕ→ℕ, f(x)=x³+2 is injective but not onto. Reason: For every natural y there is no natural…
CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Q30
Is f:ℝ∖{3}→ℝ∖{1}, f(x)=(x−2)/(x−3), one-one and onto? OR For a fixed positive integer n, determine…
📝 Chapterwise Practice Questions — Relations And Functions (227 Questions)
Class 12 Relations And Functions · Mcq Single Correct
Given the relation \(R=\{(1,2),(2,3)\}\) on the set \(A=\{1,2,3\}\), the minimum number of ordered…
Class 12 Relations And Functions · Mcq Single Correct
The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x)=e^x\) is
Class 12 Relations And Functions · Mcq Single Correct
If \(f:\mathbb{R}\to\mathbb{R}\)…
Class 12 Relations And Functions · Mcq Single Correct
If \(f:\mathbb{R}\to S\), defined by \(f(x)=\sin x-\sqrt{3}\cos x+1\), is onto, then the interval of…
Class 12 Relations And Functions · Mcq Single Correct
Let \(A=\{1,2,3\}\) and \(B=\{a,b,c\}\). Then the number of bijective functions from \(A\) to \(B\) is
Class 12 Relations And Functions · Mcq Single Correct
Let \(A\) and \(B\) be two finite sets having \(m\) and \(n\) elements respectively. Then the total…
Class 12 Relations And Functions · Mcq Single Correct
Let \(f:\mathbb{R}\to\mathbb{R}\) be defined as \(f(x)=x^4\). Then
Class 12 Relations And Functions · Mcq Single Correct
Let \(R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}\) be a relation on the set \(A=\{1,2,3,4\}\). The relation…
Class 12 Relations And Functions · Mcq Single Correct
Let \(A\) be a finite set with \(n\ge2\) elements. Which bound on the number of ordered pairs in \(R\)…
Class 12 Relations And Functions · Mcq Single Correct
For which real values of \(p\) is the function \(f:\mathbb{R}\to\mathbb{R}\), defined by \(f(x)=px+\sin…
Class 12 Relations And Functions · Mcq Single Correct
Let \(E=\{1,2,3,4\}\) and \(F=\{1,2\}\). Then the number of onto functions from \(E\) to \(F\) is
Class 12 Relations And Functions · Mcq Single Correct
Let \(R\) be the relation in the set \(\{1,2,3,4\}\) given by…
Class 12 Relations And Functions · Mcq Single Correct
Set \(A\) has 3 elements and set \(B\) has 4 elements. The number of injections that can be defined…
Class 12 Relations And Functions · Mcq Single Correct
Which of the functions defined below is one-one?
Class 12 Relations And Functions · Mcq Single Correct
Let \(L\) denote the set of all straight lines in a plane. Let a relation \(R\) be defined by \(\alpha…
Class 12 Relations And Functions · Mcq Single Correct
Let \(W\) denote the words in the English dictionary. Define \(R=\{(x,y)\in W\times W:\text{the words…
Class 12 Relations And Functions · Mcq Single Correct
For \(A=\{1,2,3\}\), define \(R=\{(1,1),(2,2),(3,3),(1,3)\}\). The ordered pair to be added to \(R\) to…
Class 12 Relations And Functions · Mcq Single Correct
Which of the following defines a one-one function from \(\mathbb{R}\) to \(\mathbb{R}\)?
Class 12 Relations And Functions · Mcq Single Correct
Let \(P=\{(x,y):|x^2+y^2|=1,\ x,y\in\mathbb{R}\}\). Then \(P\) is
Class 12 Relations And Functions · Mcq Single Correct
If \(f:\mathbb{R}\to\mathbb{C}\) is defined by \(f(x)=e^{2ix}\) for \(x\in\mathbb{R}\), then \(f\) is…
Class 12 Relations And Functions · Mcq Single Correct
Let \(R\) be the relation in \(\mathbb{N}\) given by \(R=\{(a,b):a=b-2,\ b>6\}\). Choose the correct…
Class 12 Relations And Functions · Mcq Single Correct
Let \(A=\{1,2,3\}\). Find the number of relations containing \((1,2)\) and \((1,3)\) which are…
Class 12 Relations And Functions · Mcq Single Correct
A function \(f\) from natural numbers to integers is defined by \(f(n)=\begin{cases}(n-1)/2,&n\text{…
Class 12 Relations And Functions · Mcq Single Correct
Let a relation \(R\) in \(\mathbb{N}\) be defined by \(xRy\iff x^2-4xy+3y^2=0\). The relation \(R\) is