Matching Items MCQs for Sub-Topics of Topic 9: Sets, Relations & Functions

(i) $\{x : x \text{ is a prime number less than 10}\}$

(ii) $\{x : x^2 = 16, x \in \mathbb{Z}\}$

(iii) $\{x : x \text{ is a letter in the word 'INDIA'}\}$

(iv) $\{x : x \in \mathbb{N}, x \leq 3\}$

(v) $\{x : x \text{ is an odd integer between 4 and 10}\}$

(a) $\{-4, 4\}$

(b) $\{I, N, D, A\}$

(c) $\{1, 2, 3\}$

(d) $\{5, 7, 9\}$

(e) $\{2, 3, 5, 7\}$

(i) $\mathbb{N}$

(ii) $\mathbb{Z}$

(iii) $\mathbb{Q}$

(iv) $\mathbb{R}$

(v) $\mathbb{C}$

(a) Set of rational numbers

(b) Set of complex numbers

(c) Set of integers

(d) Set of natural numbers

(e) Set of real numbers

(i) $A = \{1\}$, $B=\{a\}$, $A \times B$ is

(ii) $A = \{1, 2\}$, $B=\{a\}$, $n(A \times B)$ is

(iii) $A = \{1, 2\}$, $A \times A$ has elements

(iv) $(x, y) = (3, 5)$ implies

(v) A relation from A to B is a subset of

(a) $x=3$ and $y=5$

(b) 4

(c) $A \times B$

(d) 2

(e) $\{(1, a)\}$

(i) A well-defined collection of objects

(ii) Symbol for 'is an element of'

(iii) Representation listing elements

(iv) Representation using a property

(v) The elements $a$ and $b$ in $(a, b)$

(a) Roster form

(b) Ordered Pair Components

(c) Set

(d) $\in$

(e) Set-builder form

(i) Equality condition for $(a, b) = (c, d)$

(ii) Number of elements in $A \times B$ if $n(A)=p, n(B)=q$

(iii) Example of an ordered pair

(iv) Set of all ordered pairs $(a, b)$ from A to B

(v) Order matters in

(a) $(5, \textsf{₹}100)$

(b) Cartesian Product $A \times B$

(c) $a=c$ and $b=d$

(d) Ordered Pair

(e) $pq$

(i) Empty Set

(ii) Singleton Set

(iii) Finite Set

(iv) Infinite Set

(v) Equal Sets

(a) Contains only one element

(b) Has a definite number of elements

(c) Have exactly the same elements

(d) Contains no elements

(e) Does not have a definite number of elements

(i) $\{a, b, c\}$

(ii) $\{x : x \in \mathbb{N}, x < 5\}$

(iii) $\{0\}$

(iv) $\phi$

(v) $\{x : x^2 = 9, x \in \mathbb{R}\}$

(a) 2

(b) 0

(c) 1

(d) 3

(e) 4

(i) Proper Subset

(ii) Superset of A

(iii) Power Set of A

(iv) Universal Set

(v) Cardinality of A

(a) $P(A)$

(b) $n(A)$

(c) B if $A \subseteq B$

(d) $\subset$

(e) U

(i) $[a, b]$

(ii) $(a, b)$

(iii) $[a, b)$

(iv) $(a, b]$

(v) $[a, \infty)$

(a) $\{x : a < x \leq b\}$

(b) $\{x : a \leq x < b\}$

(c) $\{x : x \geq a\}$

(d) $\{x : a \leq x \leq b\}$

(e) $\{x : a < x < b\}$

(i) Set of all lines in a plane

(ii) Set of solutions to $x^2+1=0$ in $\mathbb{R}$

(iii) $\{1, 2, 2, 3\}$

(iv) Power set of $\phi$

(v) $\{x : x \in \mathbb{N}, x^2 < 5\}$

(a) Cardinality is 2

(b) Infinite Set

(c) Empty Set

(d) Singleton Set

(e) Cardinality is 3

(i) $\subset$

(ii) $\subseteq$

(iii) $=$

(iv) $\supset$

(v) $\supseteq$

(a) Is a superset of (could be equal)

(b) Is a proper subset of

(c) Is equal to

(d) Is a subset of (could be equal)

(e) Is a proper superset of

(i) Domain of R from A to B

(ii) Range of R from A to B

(iii) Codomain of R from A to B

(iv) A relation R from A to B is a subset of

(v) An ordered pair $(a, b) \in R$ means

(a) $A \times B$

(b) $\{x \in A : (x, y) \in R \text{ for some } y \in B\}$

(c) $\{y \in B : (x, y) \in R \text{ for some } x \in A\}$

(d) B

(e) $a$ is related to $b$ by R

(i) $R = \{(x, y) : x \text{ divides } y\}$

(ii) $R = \{(x, y) : x = y\}$

(iii) $R = \{(x, y) : x < y\}$

(iv) $R = \{(x, y) : x+y = 4\}$

(v) Universal relation

(a) $\{(1, 1), (2, 2), (3, 3)\}$

(b) $A \times A$

(c) $\{(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)\}$

(d) $\{(1, 3), (2, 2), (3, 1)\}$

(e) $\{(1, 2), (1, 3), (2, 3)\}$

(i) Elements in A from which arrows originate in an arrow diagram

(ii) Elements in A which are the first components of ordered pairs in R

(iii) Elements in A which are the second components of ordered pairs in R

(iv) Domain of R

(v) Range of R

(a) $\{a, b, c\}$ (All possible second components from A)

(b) $\{a, b, c\}$ (All possible first components from A)

(c) $\{a, b\}$

(d) $\{a, b\}$

(e) $\{a, b, c\}$

(i) Number of relations from A to B if $n(A)=2, n(B)=3$

(ii) Number of elements in $A \times B$ if $n(A)=2, n(B)=3$

(iii) Number of subsets of $A \times B$ if $n(A \times B) = 6$

(iv) Number of possible ordered pairs $(x, y)$ if $x \in \{1, 2\}, y \in \{a, b, c\}$

(v) Number of relations on A if $n(A)=2$

(a) $2^{2 \times 3} = 2^6 = 64$

(b) $2 \times 2 = 4$ relations (on A)

(c) $2 \times 3 = 6$

(d) $2^{6}$

(e) 6

(i) Reflexive

(ii) Symmetric

(iii) Transitive

(iv) Identity relation

(v) Universal relation

(a) If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$

(b) $R = A \times A$

(c) $(a, a) \in R$ for all $a \in A$

(d) If $(a, b) \in R$, then $(b, a) \in R$

(e) $R = \{(a, a) : a \in A\}$

(i) $T_1 R T_2$ if $T_1$ is congruent to $T_2$

(ii) $T_1 R T_2$ if $T_1$ has more area than $T_2$

(iii) $T_1 R T_2$ if $T_1$ is perpendicular to $T_2$

(iv) $T_1 R T_2$ if $T_1$ is similar to $T_2$

(v) $T_1 R T_2$ if $T_1$ and $T_2$ have the same perimeter

(a) Symmetric only

(b) Transitive only

(c) Equivalence relation

(d) Neither reflexive, symmetric, nor transitive

(e) Equivalence relation

(i) R is

(ii) Equivalence class of 0

(iii) Equivalence class of 1

(iv) Equivalence class of 2

(v) Equivalence class of 3

(a) Equivalence relation

(b) $\{..., -2, 1, 4, ...\}$

(c) $\{..., -1, 2, 5, ...\}$

(d) $\{..., -3, 0, 3, 6, ...\}$

(e) $\{..., -3, 0, 3, 6, ...\}$

(i) $R_1 = \{(1, 1)\}$

(ii) $R_2 = \{(1, 2), (2, 1)\}$

(iii) $R_3 = \{(1, 1), (2, 2), (1, 2), (2, 1)\}$

(iv) $R_4 = \{(1, 1), (2, 2)\}$

(v) $R_5 = \{(1, 2)\}$

(a) Reflexive, Symmetric, Transitive (Equivalence)

(b) Neither Reflexive, Symmetric, nor Transitive

(c) Symmetric only

(d) Reflexive, Symmetric, Transitive (Identity)

(e) Symmetric, Transitive (vacuously)

(i) $R = \phi$ on a non-empty set A

(ii) $R = A \times A$ on a non-empty set A

(iii) $R = \{(a, a) : a \in A\}$ on set A

(iv) R is reflexive, symmetric, and transitive

(v) Set of all elements related to a given element $a$ by an equivalence relation

(a) Identity Relation

(b) Empty Relation

(c) Equivalence Class

(d) Universal Relation

(e) Equivalence Relation

(i) $A \cup B$

(ii) $A \cap B$

(iii) $A - B$

(iv) $A'$ (Complement)

(v) Symmetric Difference $A \Delta B$

(a) $\{x : x \in A \text{ and } x \notin B\}$

(b) $(A - B) \cup (B - A)$

(c) $\{x : x \in U \text{ and } x \notin A\}$

(d) $\{x : x \in A \text{ or } x \in B\}$

(e) $\{x : x \in A \text{ and } x \in B\}$

(i) $A \cup B$

(ii) $A \cap B$

(iii) $A - B$

(iv) $B - A$

(v) $A'$

(a) $\{4, 5\}$

(b) $\{1, 2, 3, 4\}$

(c) $\{1, 2\}$

(d) $\{3\}$

(e) $\{4\}$

(i) Elements only in A

(ii) Elements in both A and B

(iii) Elements in A or B or both

(iv) Elements outside A

(v) Elements outside both A and B

(a) $A \cap B$

(b) $(A \cup B)'$

(c) $A \cup B$

(d) $A'$

(e) $A - B$

(i) $A \cup \phi$

(ii) $A \cap U$

(iii) $A \cap A'$

(iv) $(A')'$

(v) $U'$

(a) $\phi$

(b) U

(c) A

(d) $\phi$

(e) A

(i) Disjoint Sets

(ii) $A \subseteq B$

(iii) $A \subseteq A \cup B$

(iv) $A \cap B \subseteq A$

(v) $A - B$ and $B - A$

(a) Always true

(b) Mutually disjoint

(c) $A \cap B = \phi$

(d) $A \cup B = B$

(e) Always true

(i) $A \cup B = B \cup A$

(ii) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

(iii) $(A \cup B)' = A' \cap B'$

(iv) $A \cup A = A$

(v) $A \cup \phi = A$

(a) De Morgan's Law

(b) Idempotent Law

(c) Commutative Law

(d) Identity Law

(e) Distributive Law

(i) $n(A \cup B)$

(ii) $n(A \cap B)$

(iii) $n(A - B)$

(iv) $n(A \Delta B)$

(v) $n((A \cup B)')$

(a) $n(A) - n(A \cap B)$

(b) $n(A) + n(B) - n(A \cup B)$

(c) $n(A) + n(B) - 2n(A \cap B)$

(d) $n(U) - n(A \cup B)$

(e) $n(A) + n(B) - n(A \cap B)$

(i) People who like only item A

(ii) People who like A and B

(iii) People who like A or B or both

(iv) People who like neither A nor B

(v) People who like only item B

(a) $A \cap B$

(b) $A \cup B$

(c) $B - A$

(d) $(A \cup B)'$

(e) $A - B$

(i) $n(A \cup B)$

(ii) $n(A - B)$

(iii) $n(B - A)$

(iv) $n(A \Delta B)$

(v) If $n(U)=50$, $n((A \cup B)')$ is

(a) 10

(b) 30

(c) 20

(d) 25

(e) 15

(i) $A \cup A'$

(ii) $A \cap A'$

(iii) $(A')'$

(iv) $U'$

(v) $\phi'$

(a) $\phi$

(b) U

(c) A

(d) U

(e) $\phi$

(i) The set of all inputs for a function

(ii) The set of all possible outputs for a function (the target set)

(iii) The set of all actual outputs of a function

(iv) A special type of relation where each input has exactly one output

(v) The output value $f(x)$ for a given input $x$

(a) Function

(b) Codomain

(c) Range

(d) Domain

(e) Image

(i) $R_1 = \{(1, a), (2, b), (3, c)\}$

(ii) $R_2 = \{(1, a), (2, b), (1, c), (3, d)\}$

(iii) $R_3 = \{(1, a), (2, b)\}$

(iv) $R_4 = \{(1, a), (2, a), (3, a)\}$

(v) $R_5 = \{(1, a), (2, b), (3, d)\}$

(a) Not a function (element in domain has two images)

(b) Function

(c) Not a function (element in domain has no image)

(d) Function

(e) Function

(i) $f: \{1, 2, 3\} \to \mathbb{N}, f(x) = x+5$

(ii) $f: \mathbb{Z} \to \mathbb{Z}, f(x) = 2x$

(iii) $f: \mathbb{R} \to \mathbb{R}, f(x) = x^2$

(iv) $f: \mathbb{R} \to \mathbb{R}, f(x) = 7$

(v) $f: \mathbb{N} \to \mathbb{N}, f(x) = \text{largest prime factor of } x$

(a) Set of positive integers greater than 1

(b) $\{6, 7, 8\}$

(c) $\{y \in \mathbb{R} : y \geq 0\}$

(d) $\{..., -4, -2, 0, 2, 4, ...\}$

(e) $\{7\}$

(i) $f(x) = \frac{1}{x}$ (real function)

(ii) $f(x) = \sqrt{x-2}$ (real function)

(iii) $f(x) = x^3 + 2x - 1$ (polynomial)

(iv) $f(x) = \frac{x-1}{x-1}$ (real function)

(v) $f(x) = \sqrt{4-x^2}$ (real function)

(a) $\mathbb{R}$

(b) $\mathbb{R} - \{0\}$

(c) $[2, \infty)$

(d) $[-2, 2]$

(e) $\mathbb{R} - \{1\}$

(i) For $(a, b) \in f$, 'a' is the

(ii) For $(a, b) \in f$, 'b' is the

(iii) The set of all first elements of ordered pairs in a function

(iv) The set of all second elements of ordered pairs in a function

(v) The set B in $f: A \to B$

(a) Range

(b) Codomain

(c) Domain

(d) Image

(e) Pre-image

(i) Injective

(ii) Surjective

(iii) Bijective

(iv) Many-to-one

(v) Into

(a) Range is a proper subset of the codomain

(b) Different elements in domain have different images

(c) Range is equal to the codomain

(d) Both injective and surjective

(e) At least two elements in domain have the same image

(i) $f(x) = 5x - 2$

(ii) $f(x) = x^2$

(iii) $f(x) = \sin x$

(iv) $f(x) = x^3$

(v) $f(x) = |x|$

(a) Bijective

(b) Many-to-one and Into

(c) Bijective

(d) Many-to-one and Into

(e) Many-to-one and Into

(i) Injective

(ii) Surjective

(iii) Bijective

(iv) Into

(v) Many-to-one

(a) Possible

(b) Not Possible ($n(A) < n(B)$)

(c) Possible ($n(A) < n(B)$)

(d) Not Possible ($n(A) \neq n(B)$)

(e) Possible

(i) Injective

(ii) Surjective

(iii) Bijective

(iv) Into

(v) Many-to-one

(a) Possible ($n(A) > n(B)$)

(b) Not Possible ($n(A) > n(B)$)

(c) Possible

(d) Not Possible ($n(A) \neq n(B)$)

(e) Not Possible ($n(A) > n(B)$)

(i) $n(A) = n(B)$, f is injective

(ii) $n(A) > n(B)$, f is always

(iii) $n(A) < n(B)$, f is always

(iv) $n(A) = n(B)$, f is surjective

(v) $n(A) = n(B)$, f is bijective

(a) Into

(b) Bijective

(c) Onto

(d) Many-to-one

(e) Onto

(i) $f(x) = c$ (Constant function)

(ii) $f(x) = x$ (Identity function)

(iii) $f(x) = x^2$ (Square function)

(iv) $f(x) = x^3$ (Cube function)

(v) $f(x) = |x|$ (Modulus function)

(a) A V-shaped graph opening upwards from the origin

(b) A line passing through the origin with slope 1

(c) A horizontal line

(d) A parabola opening upwards from the origin

(e) A curve symmetric about the origin, passing through (0,0), (1,1), (-1,-1)

(i) $f(x) = \frac{1}{x}$ (real function)

(ii) $f(x) = \sqrt{x}$ (real function)

(iii) $f(x) = \frac{1}{x-5}$ (real function)

(iv) $f(x) = \sin x$

(v) $f(x) = \tan x$

(a) $[0, \infty)$

(b) $\mathbb{R}$

(c) $\mathbb{R} - \{x : x = (2n+1)\pi/2, n \in \mathbb{Z}\}$

(d) $\mathbb{R} - \{5\}$

(e) $\mathbb{R}$

(i) $f(x) = 2x + 3$

(ii) $f(x) = x^2 - 4$

(iii) $f(x) = |x| + 1$

(iv) $f(x) = -x^2$

(v) $f(x) = \cos x$

(a) $[-1, 1]$

(b) $[1, \infty)$

(c) $\mathbb{R}$

(d) $(-\infty, 0]$

(e) $[-4, \infty)$

(i) Graph represents a function

(ii) Graph represents a one-to-one function

(iii) Graph represents an onto function (given codomain)

(iv) X-coordinates on the graph form the domain

(v) Y-coordinates on the graph form the range

(a) Horizontal extent

(b) Every horizontal line intersects at most once

(c) Vertical line test

(d) Vertical extent

(e) Horizontal line test

(i) $f(x) = \begin{cases} x+1 & , & x < 0 \\ x^2 & , & x \geq 0 \end{cases}$, $f(-2)$ is

(ii) $f(x) = \begin{cases} x+1 & , & x < 0 \\ x^2 & , & x \geq 0 \end{cases}$, $f(3)$ is

(iii) $g(x) = \begin{cases} 2x & , & x \leq 1 \\ 3x+1 & , & x > 1 \end{cases}$, $g(1)$ is

(iv) $g(x) = \begin{cases} 2x & , & x \leq 1 \\ 3x+1 & , & x > 1 \end{cases}$, $g(2)$ is

(v) $h(x) = |x-1|$, $h(-1)$ is

(a) 9

(b) -1

(c) 7

(d) 2

(e) 2

(i) $(f+g)(x)$

(ii) $(f-g)(x)$

(iii) $(fg)(x)$

(iv) $(f/g)(x)$

(v) $(gf)(x)$

(a) $x^3 + 2x^2$

(b) $x+2+x^2$

(c) $\frac{x+2}{x^2}$

(d) $x+2-x^2$

(e) $(x+2)x^2$

(i) $(f \circ g)(x)$

(ii) $(g \circ f)(x)$

(iii) $(f \circ f)(x)$

(iv) $(g \circ g)(x)$

(v) $(f \circ g)(3)$

(a) $2x+1$

(b) 8

(c) $2x+2$

(d) $4x$

(e) $x+2$

(i) $f(x) = x^2, g(x) = \sqrt{x}$, domain of $(f+g)(x)$

(ii) $f(x) = \frac{1}{x}, g(x) = x-1$, domain of $(fg)(x)$

(iii) $f(x) = \sqrt{x}, g(x) = x-2$, domain of $(f/g)(x)$

(iv) $f(x) = \frac{1}{x-1}, g(x) = x+2$, domain of $(g \circ f)(x)$

(v) $f(x) = x^2+1$, domain of $(f \circ f)(x)$

(a) $[0, \infty) - \{2\}$

(b) $\mathbb{R} - \{1\}$

(c) $[0, \infty)$

(d) $\mathbb{R} - \{0, 1\}$

(e) $\mathbb{R}$

(i) $(f+g)(x) = (g+f)(x)$ is the ______ property.

(ii) $(f \cdot g) \cdot h = f \cdot (g \cdot h)$ is the ______ property.

(iii) $f \circ (g \circ h) = (f \circ g) \circ h$ is the ______ property.

(iv) $f \circ g = g \circ f$ is the ______ property.

(v) Domain of $(f+g)$ is $D_f \cap D_g$ is a ______ property.

(a) Not generally true (not commutative)

(b) Associative (for multiplication)

(c) Commutative (for addition)

(d) Associative (for composition)

(e) Domain property

(i) $(f \circ g)(x)$

(ii) $(g \circ f)(x)$

(iii) $(f \circ f)(x)$

(iv) $(g \circ g)(x)$

(v) $(f \circ g)(0)$

(a) $x+2$

(b) $x-2$

(c) $x$

(d) 1

(e) x

(i) Injective only

(ii) Surjective only

(iii) Bijective

(iv) Existence of $g: Y \to X$ with $f \circ g = I_Y, g \circ f = I_X$

(v) Unique image for each element in domain

(a) Condition for function

(b) Function is invertible

(c) May or may not be invertible

(d) May or may not be invertible

(e) Function is invertible

(i) $f(x) = x+5$

(ii) $f(x) = 3x$

(iii) $f(x) = x^3$

(iv) $f(x) = \frac{1}{x}$ (for $x \neq 0$)

(v) $f(x) = 2x - 1$

(a) $f^{-1}(x) = x-5$

(b) $f^{-1}(x) = \sqrt[3]{x}$

(c) $f^{-1}(x) = x/3$

(d) $f^{-1}(x) = \frac{x+1}{2}$

(e) $f^{-1}(x) = 1/x$

(i) Closure

(ii) Commutativity

(iii) Associativity

(iv) Identity element $e$

(v) Inverse element $b$ for $a$

(a) $a * b = b * a$ for all $a, b \in S$

(b) $a * e = a$ and $e * a = a$ for all $a \in S$

(c) $(a * b) * c = a * (b * c)$ for all $a, b, c \in S$

(d) $a * b \in S$ for all $a, b \in S$

(e) $a * b = e$ and $b * a = e$

(i) $\mathbb{Z}$ under addition (+)

(ii) $\mathbb{Z}$ under multiplication ($\times$)

(iii) Set of all $2 \times 2$ matrices under matrix addition

(iv) Set of all $2 \times 2$ matrices under matrix multiplication

(v) $\mathbb{R}^*$ (non-zero reals) under multiplication ($\times$)

(a) $\begin{bmatrix}0 & 0 \\ 0 & 0 \end{bmatrix}$

(b) 1

(c) 0

(d) $\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}$

(e) 1

(i) $\mathbb{Z}$ under addition (+)

(ii) $\mathbb{N}$ under subtraction (-)

(iii) $\mathbb{R}$ under $a * b = \max(a, b)$

(iv) $\mathbb{R}$ under $a * b = a - b$

(v) $\mathbb{Q}$ under $a * b = \frac{a+b}{2}$

(a) Commutative and Associative

(b) Not closed on $\mathbb{N}$

(c) Commutative but not Associative

(d) Not Commutative

(e) Commutative and Associative (e.g., identity 0)