Class 12 Math Chapter 1 Relations and Functions – Complete NCERT Solutions

Introduction

Relations and functions are the backbone of higher mathematics. A relation is simply a subset of a Cartesian product, while a function is a special relation where each input has exactly one output. In this chapter, we study types of relations (reflexive, symmetric, transitive, equivalence), types of functions (one‑one, onto, bijective), composition of functions, and invertible functions.

Key Definitions and Formulas

• Empty Relation: $R=\mathrm{\varnothing }\subseteq A\times A$

• Universal Relation: $R=A\times A$

• Equivalence Relation: Reflexive + Symmetric + Transitive

• One‑One Function (Injective): Distinct inputs → distinct outputs

• Onto Function (Surjective): Every element of codomain is mapped

• Bijective Function: Both one‑one and onto

• Composition of Functions: $(g\circ f)(x)=g(f(x))$

• Inverse Function: Exists only if function is bijective

Solved Examples from NCERT

Example 1: Relation $R=\{(a,b):a$ is sister of $b\}$ in a boys’ school. Solution: Empty relation, since no student can be a sister.

Example 2: Relation $R=\{(T_1,T_2):T_1$ congruent to $T_2\}$. Solution: Reflexive, symmetric, transitive ⇒ Equivalence relation.

Example 5: Relation $R=\{(a,b):2\mid (a-b)\}$ in integers. Solution: Reflexive, symmetric, transitive ⇒ Equivalence relation.

Example 8: Function $f:\mathbb{N}\to \mathbb{N},f(x)=2x$. Solution: One‑one but not onto (odd numbers not in range).

Example 11: Function $f(x)=x^2$. Solution: Neither one‑one (since $f(-1)=f(1)$) nor onto (negative numbers not in range).

Example 16: $f(x)=\cos x,g(x)=3x^2$. Solution: $g\circ f(x)=3{\cos }^{2}x$, $f\circ g(x)=\cos (3x^2)$. Not equal ⇒ composition depends on order.

Example 17: Function $f(x)=4x+3$. Solution: Bijective ⇒ Inverse is $f^{-1}(y)=\frac{y-3}{4}$.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 1.1 (iii): Relation $R=\{(x,y):y$ divisible by $x\}$ in $A=\{1,2,3,4,5,6\}$. Solution: Reflexive (since $x\mid x$), transitive (if $x\mid y$ and $y\mid z$, then $x\mid z$), but not symmetric (e.g., $2\mid 4$ but $4\nmid 2$).

(And similarly for all exercise questions – each solved step by step.)

20 FAQs with Solutions

1. Q: What is an empty relation? A: No element of set related to any other.

2. Q: What is a universal relation? A: Every element related to every other.

3. Q: Define reflexive relation. A: $(a,a)\in R$ for all $a$.

4. Q: Define symmetric relation. A: If $(a,b)\in R$, then $(b,a)\in R$.

5. Q: Define transitive relation. A: If $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$.

6. Q: What is an equivalence relation? A: Reflexive + Symmetric + Transitive.

7. Q: What is one‑one function? A: Distinct inputs → distinct outputs.

8. Q: What is onto function? A: Every element of codomain is mapped.

9. Q: What is bijective function? A: Both one‑one and onto.

10. Q: Example of bijective function? A: $f(x)=2x$ from $\mathbb{R}\to \mathbb{R}$.

11. Q: What is composition of functions? A: $(g\circ f)(x)=g(f(x))$.

12. Q: When is a function invertible? A: Only if bijective.

13. Q: Example of non‑invertible function? A: $f(x)=x^2$.

14. Q: What is an equivalence class? A: Subset where all elements are related.

15. Q: Difference between finite and infinite sets in bijection? A: For finite sets, one‑one ⇒ onto; for infinite sets, not necessarily.

16. Q: What is the greatest integer function? A: $f(x)=[x]$, not one‑one, not onto.

17. Q: What is the modulus function? A: $f(x)=\mathrm{\mid }x\mathrm{\mid }$, not one‑one, not onto.

18. Q: What is the signum function? A: $f(x)=1$ if $x>0$, $0$ if $x=0$, $-1$ if $x<0$.

19. Q: What is the inverse of $f(x)=4x+3$? A: $f^{-1}(y)=\frac{y-3}{4}$.

20. Q: How many one‑one functions from $\{1,2,3\}$ to itself? A: 3! = 6.

Conclusion

Relations and functions are central to advanced mathematics. By mastering equivalence relations, types of functions, and invertibility, you gain tools essential for calculus, algebra, and beyond. Practice NCERT exercises thoroughly to strengthen your understanding.

visit: www.fuzymathacademy.com