# SSC > Set Theory

Explore popular questions from Set Theory for SSC. This collection covers Set Theory previous year SSC questions hand picked by experienced teachers.

General Intelligence and Reasoning
General Awareness
Quantitative Aptitude
English Comprehension
Q 1.

Correct2

Incorrect-0.5

{tex} A- \left(B \cap C\right) {/tex} is equal to

A

{tex} \left(A-B\right)\cap \left(A-C\right) {/tex}

B

{tex} \left(A-B\right)\cup \left(A-C\right) {/tex}

C

{tex} \left(A\cap B\right)-C {/tex}

None of these

##### Explanation

From Venn diagram

{tex} \left(A \cap B\right)^{c} {/tex} = Portion exterior to

{tex} \left(A \cap B\right)^{c} \cap A {/tex} = Portion showing both shadings = A-B

Q 2.

Correct2

Incorrect-0.5

If A = {tex} \{ x : x^{2} = 1 \} {/tex} and B {tex} \{ x : x^{4} = 1 \} {/tex}, then {tex} A \cup B {/tex} is equal to:

{i, -i}

B

{-1, 1}

C

{-1, 1, i. -i}

D

None of these

##### Explanation

Given that {tex} A = \{ x:x^{2}=1 \},\ B=\{ x:x^{4}=1 \} {/tex}

=> {tex} A = \{ -1,\ 1 \},\ B = \{ -1,\ 1,\ -i,\ i \} {/tex}

Now, {tex} A - B = \phi,\ B-A = \{ -i,\ i \} {/tex}

Therefore, {tex} \left(A-B\right) \cup \left(B-A\right) = \{ -i,\ i \} {/tex}

Q 3.

Correct2

Incorrect-0.5

If A = {tex} \{ x : x = 4n+1, 2 \le n \le 5 \} {/tex} then number of subsets of A is:

A

16

15

C

4

D

none of these

##### Explanation

Given that {tex} A = \{ x:x=4n+1;\ 2 \le n \le 5 \} {/tex}

Number of elements in set A is 4 ,

So, number of proper subsets = {tex} 2^{4} - 1 = 15 {/tex}.

Q 4.

Correct2

Incorrect-0.5

Let R and S be two relations on a set A. Then which is not correct?

R and S are transitive, then R u S is also transitive.

B

R and S are transitive, then R n S is also transitive.

C

R and S are reflexive, then R n S is also reflexive.

D

R and S are symmetric, then R U S is also symmetric

##### Explanation

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Q 5.

Correct2

Incorrect-0.5

The group of beautiful girls is:

A

a null set

B

a finite set

C

a singleton set

not a set

##### Explanation

Beautiful is relative term so, it is not well defined term. Therefore, it is not a set.

Q 6.

Correct2

Incorrect-0.5

R is a relation over the set of real numbers and it is given by {tex} nm \ge 0 {/tex}. Then R is:

A

symmetric and transitive

B

reflexive and symmetric

C

a partial order relation

an equivalence relation

##### Explanation

---

Q 7.

Correct2

Incorrect-0.5

In a city of 55 students, the number of students studying different subjects are 23 in mathematics, 24 in physics, 19 in chemistry, 12 in mathematics and physics, 9 in mathematics and chemistry, 7 in physics and chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is:

A

6

B

9

7

D

all of these

##### Explanation

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Q 8.

Correct2

Incorrect-0.5

If {tex} N_{a}=\{ an : n \epsilon N \} {/tex}, then {tex} N_{3} \cap N_{4} {/tex} is equal to:

A

{tex} N_{7} {/tex}

{tex} N_{12} {/tex}

C

{tex} N_{3} {/tex}

D

{tex} N_{4} {/tex}

##### Explanation

Given that {tex} N_{a}=\{ an:n \epsilon N \} {/tex}

{tex} N_{3} \cap N_{4} = \{ 3,\ 6,\ 9.\ 12,\ 15, ..... \} \cap \{ 4,\ 8,\ 12,\ 16,\ 20, ... \} {/tex}

= {tex} \{ 12,\ 24,\ 36, ... \} = N_{12} {/tex}

Q 9.

Correct2

Incorrect-0.5

The relation "Congruence modulo m" is:

A

reflexive only

B

transitive only

C

symmetric only

an equivalence relation

##### Explanation

{tex} x=3 \left(mod\ 7\right) => x-3 = 7p,\ \left(p\ \epsilon\ I\right) {/tex}

=> {tex} x=7p+3,\ p\ \epsilon\ I\ i.e.,\ \{ 7p+3 : p\ \epsilon\ z \} {/tex}

Therefore, Solution set of x is {tex} \{ 7p + 3 : p\ \epsilon\ I \} {/tex}.

Q 10.

Correct2

Incorrect-0.5

Set A has 3 elements and set B has 4 elements. The number of injections that can be defined from A to B is:

A

144

B

12

24

D

30

##### Explanation

Given that {tex} n \left(A\right)=3 {/tex} and {tex} n \left(B\right)=4 {/tex}, the number of injections or one-one mapping is given by.

{tex} ^{4}p_{3} \frac{4 !}{ \left(4-3\right)!}= 4 \times 3 \times 2 \times 1 = 24 {/tex}

Q 11.

Correct2

Incorrect-0.5

Which of the four statements given below is different from the other?

A

{tex} f:A \rightarrow B {/tex}

{tex} f:x \rightarrow f \left(x\right) {/tex}

C

f is a mapping from A to B

D

f is a function from A to B

##### Explanation

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Q 12.

Correct2

Incorrect-0.5

Which of the following is correct?

A

{tex} A \cap B \subset A \cup B {/tex}

{tex} A \cap B \subseteq A \cup B {/tex}

C

{tex} A \cup B \subset A \cap B {/tex}

D

None of these

##### Explanation

If {tex} A = B,\ then\ A\cap B = A \cup B {/tex}

{tex} A \cap B \subseteq A \cup B {/tex}

Q 13.

Correct2

Incorrect-0.5

Let {tex} f:N \rightarrow R:f \left(x\right)=\frac{ \left(2x-1\right) }{2} {/tex} and {tex} g:Q \rightarrow R:g \left(x\right)=x+2 {/tex} be two functions then {tex} \left(gof\right) \left(\frac{3}{2}\right) {/tex}

3

B

1

C

{tex} \frac{7}{2} {/tex}

D

None of these

##### Explanation

{tex} f \left(x\right)=\frac{2x-1}{2}\ and\ g \left(x\right)=x+2 {/tex}

{tex} \left(gof \left(x\right) \right)=g \left(\frac{1}{2} \left(2x-1\right) \right) {/tex}

= {tex} x-\frac{1}{2}+2 = x+\frac{3}{2} {/tex}

{tex} \therefore \left(gof\right) \left(\frac{3}2{}\right)=\frac{3}{2}+\frac{3}{2}=3 {/tex}

But, {tex} \frac{3}{2} \notin N{/tex}

Q 14.

Correct2

Incorrect-0.5

If N be the set of all natural numbers, consider {tex} f:N \rightarrow N:f \left(x\right)=2x \forall x \epsilon N {/tex}, then f is:

A

one-one onto

one-one into

C

many-one

D

one of these

##### Explanation

Let {tex} x_{1},x_{2} \epsilon N {/tex}, then {tex} f \left(x_{1}\right)=f \left(x_{2}\right) {/tex}

=> {tex} 2x_{1}=2x_{2} => x_{1}=x_{2} {/tex}

Let y = 2x, then {tex} x = \frac{y}{2} \notin N {/tex}

Now, if we put y = 5, then {tex} x = \frac{5}{2} \notin N {/tex}

This shows that {tex} 5 \epsilon N {/tex}, has no pre image in n

So, f is into

Hence, f is one-one into

Q 15.

Correct2

Incorrect-0.5

N is the set of natural numbers. The relation R is defined on {tex} N \times N {/tex} as follows: {tex} \left(a,\ b\right)R \left(c,\ d\right) \Leftrightarrow a+d=b+c {/tex} is:

A

reflexive

B

symmetric

C

transitive

all of these

##### Explanation

(a) {tex} \left(a,\ b\right) R \left(a,\ b\right) \Leftrightarrow \ a+b=b+a {/tex}

Therefore, R is reflexive

(b) {tex} \left(a,\ b\right) R \left(c,\ d\right) => a+d = b+c {/tex}

=> {tex} c+b = d+a {/tex}

=> {tex} \left(c,\ d\right) R \left(a,\ b\right) {/tex}

Therefore, R is symmetric

(c) {tex} \left(a,\ b \right) R \left(c,\ d\right)\ and\ \left(c,\ d\right) R \left(e,\ f \right) {/tex}

=> {tex} a+d=b+c\ and \ c + f=d+e {/tex}

=> {tex} a+b+c+f = b+c+d+e {/tex}

=> {tex} a+f = b+e {/tex}

=> {tex} \left(a,\ b\right) R \left(e,\ f\right) {/tex}

Therefore, R is transitive

Thus R is an equivalence relation {tex} N \times N {/tex}

Q 16.

Correct2

Incorrect-0.5

If {tex} E=\{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}; {/tex} {tex} A=\{ 1, 3, 5, 7 \},\ B=\{ 2, 4, 6, 8 \} {/tex} then find the value of {tex} \left(A\cup B\right)' {/tex} and {tex} A'\cap B' {/tex}

A

{tex} \{ 6 \} {/tex}

{tex} \{ 9 \} {/tex}

C

{tex} \{ \} {/tex}

D

None of these

##### Explanation

{tex} E = \{ 1,2,3,4,5,6,7,8,9,10 \} {/tex}

{tex} A = \{ 1,3,5,7 \} {/tex}

{tex} B = \{ 2,4,6,8,10 \} {/tex}

{tex} A\cup B = \{ 1,2,3,4,5,6,7,8,9,10 \} {/tex}

{tex} \left(A\cup B\right)' = \{ 9 \} {/tex}

{tex} A' = \{ 2,4,6,8,9,10 \} {/tex}

{tex} B' = \{ 1,3,5,7,9 \} {/tex}

{tex} A'\cap B' = \{ 9 \} {/tex}

Q 17.

Correct2

Incorrect-0.5

{tex} A=\{ 4, 9, 16, 25, 36 \},\ B=\{ 9, 25, 49, 81 \},\ C=\{ 16, 81, 256 \} {/tex} then find {tex} A- \left(B\cap C\right) {/tex}

A

{tex} \{ 81 \} {/tex}

{tex} \{ 4, 9, 16, 25, 36 \} {/tex}

C

{tex} \{ 4, 36 \} {/tex}

D

{tex} \{ 4, 16, 36 \} {/tex}

##### Explanation

{tex} A=\{ 4,\ 9,\ 16,\ 25,\ 36 \},\ B=\{ 9,\ 25,\ 49,\ 81 \},\ C=\{ 16,\ 81,\ 256 \} {/tex}

{tex} B\cap C = \{ 81 \} {/tex}

{tex} A- \left(B\cap C\right) = \{ 4,\ 9,\ 16,\ 25,\ 36 \}-\{ 81 \} {/tex}

={tex} \{ 4,\ 9,\ 16,\ 25,\ 36 \} {/tex}

Q 18.

Correct2

Incorrect-0.5

If {tex} n \left(A\right)=6, n \left(B\right)=5, n \left(A\cap B\right)=3 {/tex} then find {tex} n \left(A\cup B\right) {/tex}

A

11

B

9

8

D

None of these

##### Explanation

{tex} n \left(A\cup B\right) = n \left(A\right)+n \left(B\right)-n \left(A\cap B\right) {/tex}

= 6+5-3 = 8

Q 19.

Correct2

Incorrect-0.5

lf {tex} A=\{ 1, 2, 3, 4, 5, 6 \},\ B=\{ 4, 5, 6 \} {/tex}, then {tex} n \left(A\cap B\right) {/tex}

3

B

6

C

4

D

2

##### Explanation

{tex} A = \{ 1,2,3,4,5,6 \} {/tex}

{tex} B = \{ 4,5,6 \} {/tex}

{tex} A\cap B = \{ 4,5,6 \} {/tex}

{tex} n \left(A\cap B\right) = 3 {/tex}

Q 20.

Correct2

Incorrect-0.5

If {tex} n \left(p \left(A\right)=16 \right) {/tex} then what is the value of {tex} n \left(A\right) {/tex}

A

12

4

C

8

D

2

##### Explanation

{tex} nP \left(A\right) = 16 = 2^{4} {/tex}

The value of {tex} n \left(A\right) = 4 {/tex}

Q 21.

Correct2

Incorrect-0.5

If {tex} A=\{ 4, 5, 6, 7, 8, 9, 10 \},\ A\cap B=\{ 6, 7, 8 \},\ A\cup B=\{ 4, 5, 6, 7, 8, 9, 10, 11, 12 \} {/tex}, then find B.

A

{tex} \{ 4, 5, 6, 7, 8 \} {/tex}

{tex} \{ 6, 7, 8, 11, 12 \} {/tex}

C

{tex} \{ 4, 5, 7, 10, 11, 12 \} {/tex}

D

None of these

##### Explanation

{tex} A \cup B = \{ 4,5,6,7,8,9,10,11,12 \} {/tex}

{tex} A = \{ 4,5,67,8,9,10 \} {/tex}

{tex} A\cap B = \{ 6,7,8 \} {/tex}

{tex} B = \{ 6,7,8,11,12 \} {/tex}

Q 22.

Correct2

Incorrect-0.5

If {tex} E=\{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \},\ A=\{1, 2, 3, 4, 5 \},\ B=\{2, 4, 6, \} {/tex} then find {tex} \left(A\cup B\right)' {/tex}

{tex} \{ 8, 9 10 \} {/tex}

B

{tex} \{ 7, 8, 9 10 \} {/tex}

C

{tex} \{ 6, 7, 8, 9, 10 \} {/tex}

D

cannot be found

##### Explanation

{tex} E = \{ 1,2,3,4,5,6,7,8,9,10 \} {/tex}

{tex} A = \{ 1,2,3,5 \},\ B = \{ 2,4,6,7 \} {/tex}

{tex} A\cup B = \{ 1,2,3,4,5,6,7 \} {/tex}

{tex} \left(A\cup B\right)' = \{ 8,9,10 \} {/tex}

Q 23.

Correct2

Incorrect-0.5

{tex} n \left(A\cup B\right)=8,\ n \left(A\right)=6,\ n \left(B\right)=4 {/tex} find {tex} n \left(A\cap B\right) {/tex}

A

10

B

7

2

D

4

##### Explanation

{tex} n \left(A\cup B\right)=8,\ n \left(A\right)=6,\ n \left(B\right)=4 {/tex}

{tex} n \left(A\cup B\right)=n \left(A\right)+n \left(B\right)-n \left(A\cap B\right) {/tex}

{tex} 8 = 6+4-n \left(A\cap B\right) {/tex}

{tex} n \left(A\cap B\right) = 10-8=2 {/tex}

Q 24.

Correct2

Incorrect-0.5

In an examination {tex} 77 ^{\circ} {/tex} pupils passed in English {tex} 65 ^{\circ} {/tex} students passed in Mathematics and {tex} 50 ^{\circ} {/tex} students passed both in English and Mathematics. Then how many percentage of students fail in both the subjects.

A

16

8

C

12

D

20

##### Explanation

Let the total number of students be 100

Passed in Maths = 77

Passed in English = 65

Passed in both subjects 50

Total Number of passed students = 27 + 50 + 15 = 92

[77 + 68 - 50]

Failed in both the subjects = 100 - 92 = 8

Q 25.

Correct2

Incorrect-0.5

In an examination, {tex} 45 ^{\circ} {/tex} students failed in Science and {tex} 56 ^{\circ} {/tex} failed in Mathematics if {tex} 16 ^{\circ} {/tex} failed in both Science and Mathematics the percentage of those who passed in both the subject is.

15

B

31

C

49

D

42

##### Explanation

Let the number of students appeared for the examination be 100.

Let the circle A represent who failed 1n Science and

B represent the students who failed in Mathematics respectively.

Number of students failed in Science only = 45 - 16 = 29.

Number of students failed in Mathematics only = 56 - 16 = 40

Total No of students failed = Number of students failed in science + Number of students failed in Mathematics + failed in both.

= 29 + 40 + 16

Number of students passed = 100 - 85 = 15