Sets, Functions & Groups

Practise Sets, Functions & Groups for FBISE Grade 9–10 Mathematics (Grade 9–10): 8 multiple-choice questions with instant checking and worked explanations. Free, no sign-up.

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1. Question 1

   Which set is a subgroup of the additive group $(\mathbb{Z},+)$?
   • A{1,2,3}
   • B{3,6,9}
   • C{0}
   • D{2,4,6}
   Show answer and explanation

   Answer: C) {0}

   A subgroup must contain the additive identity $0$, be closed under addition, and contain additive inverses. $\{0\}$ meets all three criteria, so it is a subgroup. The other sets lack $0$, so they fail the identity requirement and cannot be subgroups.

2. Question 2

   In the set $G=\{1,5,7,11\}$ with multiplication modulo $12$, let $H=\{1,5\}$. Which statement about $H$ is correct?
   • AH is a subgroup of G
   • BH contains the identity but lacks inverses
   • CH is equal to G
   • DH is not closed under the operation
   Show answer and explanation

   Answer: A) H is a subgroup of G

   H contains the identity element 1, is closed under multiplication modulo 12, and each element has an inverse within H (5·5≡1, so 5 is its own inverse). Therefore H satisfies all subgroup criteria and is a subgroup of G. The other options are wrong because closure actually holds, inverses are present, and H has fewer elements than G.

3. Question 3

   For $f:\{1,2,3,4,5,6\}\to \{a,b,c\}$ defined by $f(1)=a$, $f(2)=b$, $f(3)=c$, $f(4)=a$, $f(5)=b$, $f(6)=c$, let $S=\{2,4,5\}$. If $I=f(S)$ and $P=f^{-1}(I)$, which set equals $P$?
   • A{1,2,3,4,5,6}
   • B{2,4,5}
   • C{2,4,5,6}
   • D{1,2,4,5}
   Show answer and explanation

   Answer: D) {1,2,4,5}

   The image of S under f is {a,b}. The pre‑image of this set consists of all domain elements that map to a or b, namely 1,2,4,5. Hence P = {1,2,4,5}. The other options ignore the extra elements that share the same images or mistakenly assume the pre‑image equals S or the whole domain.

4. Question 4

   Let $A=\{2,4,6,8\}$ and $B=\{1,3,5,7,9\}$. How many ordered pairs are in $A\times B$?
   • A12
   • B24
   • C20
   • D16
   Show answer and explanation

   Answer: C) 20

   The Cartesian product $A\times B$ consists of all ordered pairs $(a,b)$ with $a\in A$ and $b\in B$. Since $A$ contains four numbers and $B$ contains five, the total number of pairs is $4\times 5=20$. Therefore the correct answer is 20. The option 12 results from mistakenly counting only three elements in $A$; 16 comes from counting four elements in $B$; 24 comes from counting six elements in $B$.

5. Question 5

   On $\{1,2,3\}$, define $xRy$ iff $x\equiv y(\mathrm{mod}2)$. The relation $R$ is:
   • AReflexive only
   • BSymmetric only
   • CAn equivalence relation
   • DNeither reflexive nor symmetric
   Show answer and explanation

   Answer: C) An equivalence relation

   The relation $xRy$ defined by $x\equiv y(\mathrm{mod}2)$ is reflexive (each element is congruent to itself), symmetric (if $x\equiv y$ then $y\equiv x$), and transitive (if $x\equiv y$ and $y\equiv z$ then $x\equiv z$). Therefore it satisfies all three properties and is an equivalence relation. The other options incorrectly omit one or more properties.

6. Question 6

   The range of $f(x)=\sqrt{x}$ (domain $x\ge 0$) is:
   • A$\mathbb{R}$
   • B$(-\infty ,0]$
   • C$[0,\infty )$
   • D$\{0\}$
   Show answer and explanation

   Answer: C) $[0,\infty )$

   The square‑root function is defined only for non‑negative inputs and always returns the non‑negative root, so its outputs are all numbers $y\ge 0$, giving the range $[0,\infty )$. $(-\infty ,0]$ is wrong because the function never produces negative values. $\mathbb{R}$ is wrong because negative numbers are not outputs. $\{0\}$ is wrong because any positive $x$ yields a positive square root, so more than just zero is possible.

7. Question 7

   Let $f:\mathbb{Z}\to {\mathbb{Z}}_8$ be defined by $f(x)=x(\mathrm{mod}8)$. Let $H=\{0,4\}$ be a subset of the additive group $({\mathbb{Z}}_8,+)$. Define $K=f^{-1}(H)$. Which statement about $K$ is true?
   • A$K=\{4k\mid k\in \mathbb{Z}\}$
   • B$K$ consists of all even integers
   • C$K$ is not closed under addition
   • D$K$ is a subgroup of $({\mathbb{Z}}_8,+)$
   Show answer and explanation

   Answer: A) $K=\{4k\mid k\in \mathbb{Z}\}$

   K consists of all integers whose remainder modulo 8 is either 0 or 4; those are exactly the multiples of 4, so K = {4k | k∈ℤ}. The option “$K=\{4k\mid k\in \mathbb{Z}\}$” states this precisely, making it the correct choice.

8. Question 8

   Let $A=\{1,2,3,4\}$ and $B=\{a,b,c\}$. A function $f:A\to B$ is defined by $f(1)=a$, $f(2)=b$, $f(3)=a$, $f(4)=c$. What is $f(\{1,3,4\})$?
   • A$\{a,c\}$
   • B$\{a,b,c\}$
   • C$\{b,c\}$
   • D$\{a,b\}$
   Show answer and explanation

   Answer: A) $\{a,c\}$

   The image of the set {1,3,4} under f consists of the outputs for those inputs: f(1)=a, f(3)=a, and f(4)=c, giving the set {a,c}. Hence the option “$\{a,c\}$” is correct.

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