Relations and Functions: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Define a symmetric relation.
2. Give an example of a relation that is transitive only.
3. Find $[3]$ for $aRb⟺4\mid (a-b)$ on $\{0,\dots ,10\}$.
4. Is $f(x)=x^2+1$, $R\to R$ onto?
5. How many one-one functions from a 2-set to a 4-set?
6. If $f(x)=x^2$, $g(x)=x+1$, find $g\circ f(2)$.
7. Write the range of the signum function.
8. When is a function invertible?
9. Find the number of relations from $\{a,b\}$ to $\{1,2,3\}$.
10. Is "$a<b$" on R reflexive?

D. Short answer I (2 marks each)

1. Show that $f(x)=7x-2$, $R\to R$ is bijective.
2. Check whether $R=\{(a,b):a-b\text{ is odd}\}$ on Z is transitive.
3. Show that $f:N\to N$, $f(x)=x+3$ is not onto.
4. Find all equivalence classes of $aRb⟺3\mid (a-b)$ on $\{1,\dots ,7\}$.
5. Show that $f(x)=\frac{x}{x+1}$, $x\ne -1$ is one-one.
6. Find $g\circ f$ and $f\circ g$ for $f(x)=|x|$, $g(x)=|5x-2|$.
7. Give a relation on $\{1,2,3\}$ that is reflexive and symmetric but not transitive.
8. Is $f:Z\to Z$, $f(x)=2x$ onto? Justify.

Rough work

E. Short answer II (3 marks each)

1. Show that $R=\{(a,b):a+b\text{ even}\}$ on Z is an equivalence relation; write $[0]$.
2. Show that $f:R-\{2\}\to R-\{1\}$, $f(x)=\frac{x-1}{x-2}$ is a bijection.
3. On $N\times N$, $(a,b)R(c,d)⟺a+d=b+c$: prove R is an equivalence relation.
4. Show that $f(x)=\frac{x}{x^2+1}$, $R\to R$ is neither one-one nor onto.
5. Let $A=\{1,2,3\}$. Write the smallest equivalence relation containing $(1,2)$ and $(2,3)$.
6. Show that $f:N\to N$, $f(n)=n+1$ ($n$ odd), $n-1$ ($n$ even) is bijective.

F. Long answer (5 marks each)

1. (a) On the set of real numbers, $aRb⟺a-b+\sqrt{3}$ is irrational. Check reflexive, symmetric, transitive. (b) Show that $f:R\to R$, $f(x)=\frac{2x-1}{3}$ is invertible and find $f^{-1}$.
2. (a) Show that $f:[-1,1]\to R$, $f(x)=\frac{x}{x+2}$ is one-one; find the range and the inverse $f:[-1,1]\to \text{Range}$. (b) How many functions, one-one functions and onto functions are there from $\{1,2,3\}$ to $\{a,b\}$?

G. HOTS (higher-order thinking)

1. Can a relation be symmetric and transitive but not reflexive? Give an example.
2. Why is every strictly increasing function one-one?
3. If $f\circ g$ is one-one, must $g$ be one-one?
4. How many reflexive relations are there on a set of 3 elements?

20 MCQs

1. Number of relations on $\{1,2\}$
   (a) 4 (b) 8 (c) 16 (d) 2
2. $\{(1,1),(2,2)\}$ on $\{1,2,3\}$ is
   (a) symmetric and transitive (b) reflexive (c) an equivalence (d) universal
3. $f(x)=2x$, $N\to N$ is
   (a) onto (b) many-one (c) bijective (d) one-one, not onto
4. $f(x)=x^2$, $R\to R$ is
   (a) one-one (b) neither one-one nor onto (c) onto (d) bijective
5. Bijections on a 3-element set
   (a) 3 (b) 6 (c) 9 (d) 27
6. "Is perpendicular to" on lines is
   (a) reflexive (b) equivalence (c) transitive (d) symmetric only
7. $aRb⟺a\le b$ on R is not
   (a) symmetric (b) reflexive (c) transitive (d) a relation
8. Equivalence classes of $aRb⟺2\mid (a-b)$ on Z
   (a) 1 (b) 3 (c) 2 (d) infinitely many
9. Number of functions from $\{1,2,3\}$ to $\{a,b\}$
   (a) 6 (b) 5 (c) 9 (d) 8
10. One-one functions from a 4-set to a 3-set
    (a) 24 (b) 0 (c) 12 (d) 81
11. $f(x)=x^3$, $R\to R$
    (a) many-one (b) into (c) bijective (d) not a function
12. If $f(x)=3x-4$, $f^{-1}(x)=$
    (a) $\frac{x+4}{3}$ (b) $3x+4$ (c) $\frac{x-4}{3}$ (d) $4-3x$
13. $g\circ f(x)$ for $f=x+1$, $g=x^2$
    (a) $x^2+1$ (b) $(x+1{)}^2$ (c) $x^2+2$ (d) $2x$
14. $[x]$ on R is
    (a) onto (b) one-one (c) neither one-one nor onto (d) bijective
15. An onto function from $\{1,2,3\}$ to itself is always
    (a) constant (b) identity (c) many-one (d) one-one
16. Equivalence relations on $\{1,2,3\}$ containing $(1,2)$
    (a) 2 (b) 1 (c) 3 (d) 4
17. $\{(1,2),(2,1)\}$ on $\{1,2\}$ is
    (a) symmetric (b) transitive (c) reflexive (d) equivalence
18. $f(x)={\log }_{2}x$, $(0,\infty )\to R$
    (a) not onto (b) into (c) many-one (d) bijective
19. Range of $f(x)=\frac{x}{1+|x|}$
    (a) R (b) $(-1,1)$ (c) $[0,1)$ (d) $[-1,1]$
20. Relations from a 2-set to a 3-set
    (a) 6 (b) 8 (c) 64 (d) 9

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: $f(x)=x^2$, $R\to R$ is not one-one. R: $f(-2)=f(2)$.
2. A: $(1,1),(2,2),(3,3)$ on $1,2,3$ is an equivalence relation. R: every equivalence relation is reflexive.
3. A: $f(x)=e^x$, $R\to R$ is not onto. R: $e^x>0$ for all $x$.
4. A: There are 8 relations on $\{1,2\}$ that are reflexive. R: reflexive relations on an $n$-set number $2^{n^2-n}$.
5. A: $f:N\to N$, $f(x)=x^2$ is one-one. R: the square of a natural number is a natural number.
6. A: Every bijection has an inverse. R: a function is invertible if and only if it is one-one and onto.

Key: 1 (a) · 2 (b) · 3 (a) · 4 (d) · 5 (b) · 6 (a)

Answers to the practice set