Mathematics Class XI

Question 1 of 25

1. Question
If $A$ and $B$ are any sets, $n(A) = 5,n(B) = 5,$ then which relation satisfy for $A$ and $B$?
- $A \subset B$
- $A = B$
- $A \sim B$
- None of these
Correct

Incorrect

Hint

$A$ and $B$ are any sets.
$n(A) = 5,n(B) = 5$
But we can’t say that $A$ and $B$ . Both have equal elements.
As $n(A) = n(B)$ , these $A$ and $B$ are said to be similar.
$\therefore A \sim B$ .
Question 2 of 25

2. Question
If $A,B$ and $C$ be any sets. Then $A \times (B \cup C)$ is equal to
- $(A \times B) \cup C$
- $(A \times B) \cup (A \times C)$
- $(A \cup B) \times C$
- None of these
Correct

Incorrect

Hint

$A \times (B \cup C) = (A \times B) \cup (A \times C)$ .
Question 3 of 25

3. Question
If $A,B$ and $C$ be sets, then $A – (B \cup C)$ is equal to
- $(A – B) \cup C$
- $A – (BC)$
- $(A – B) – C$
- None of these
Correct

Incorrect

Hint

$A,B$ and $C$ be sets
Let $x \in A - (B \cup C)$
$\Leftrightarrow x \in A$ and $[x \notin (B \cup C)]$
$\Leftrightarrow x \in A$ and $[x \notin B/latex] and [mathjax][latex]x \notin C]$
$\Leftrightarrow [x \in A/latex] and [mathjax][latex]x \notin B]$ and $x \notin C$
$\Leftrightarrow [x \in (A - B)]$ and $x \notin C$
$\Leftrightarrow x \in (A - B) - C$
$\therefore A - (B \cup C) = (A - B) - C$ .
Question 4 of 25

4. Question
If $A,B$ and $C$ be sets, then $(A \cap B) – C$ Is equal to
- $A \cap (B – C)$
- $(A – B)$
- $(A – C)$
- None of these
Correct

Incorrect

Hint

$x \in (A \cap B) - C$
$\Leftrightarrow (x \in A$ and $x \in B)$ and $x \notin C$
$\Leftrightarrow x \in A$ and $(x \in B$ and $x \notin C)$
$\Leftrightarrow x \in A$ and $(x \in (B - C))$
$\Leftrightarrow x \in A \cap (B - C)$
$\therefore (A \cap B) - C = A \cap (B - C)$ .
Question 5 of 25

5. Question
If $A,B$ and $C$ be sets. Then $A \times (B \cap C)$ is equal to
- $(A \times B) \cap C$
- $(A \times B) \cap (A \times C)$
- $(A \times B) \cap (B \times C)$
- None of the above
Correct

Incorrect

Hint

$A \times (B \cap C) = (A \times B) \cap (A \times C)$ .
Question 6 of 25

6. Question
If $A,B$ and $C$ be sets. Then is
- .$(A \cap B)\vartriangle (A \cap C)$
- .$ (A \cap B)\vartriangle C$
- $ A\vartriangle (B \cap C) $
- None of these
Correct

Incorrect

Hint

$A,B$ and $C$ be sets.
Let
$\Leftrightarrow x \in A$ and
$\Leftrightarrow x \in A$ and $(x \in B$ and $x \notin C$ or $x \in C$ and $x \notin B)$
$\Leftrightarrow (x \in A$ and $x \in B$ and $x \in C)$ or $(x \in A$ and $x \in C$ and $x \notin B)$
$\Leftrightarrow (x \in (A \cap B)$ and $x \notin (A \cap B))$ or $x \in (A \cap C)$ and $x \notin (A \cap B))$
$\Leftrightarrow (x \in (A \cap B) - (A \cap C))$ or $(x \in (A - C) - (A \cap B))$
.
Question 7 of 25

7. Question
If $A$ and $B$ two sets, then $P(A) \subset P(B)$ is related to
- $A – B$
- $A \subset B$
- $A \supset B$
- None of these
Correct

Incorrect

Hint

$A$ and $B$ be sets.
Let $n(A) = m,n(B) = n$
$P(A) \subset P(B)$
$\Leftrightarrow {2^m} < {2^n}$
$\Leftrightarrow m < n$ $(\therefore m$ and $n$ are non-negative integers $)$
$\begin{array}{l} \Leftrightarrow n(A) < n(B)\\ \Leftrightarrow A \subset B\end{array}$
$P(A) \subset P(B)$ If and only if $A \subset B$ .
Question 8 of 25

8. Question
If $A = \{ x,y\} ,$ then $A \times A$ is equal to
- $\{ (x,x),(x,y),(y,x),(y,y)\} $
- $\{ \{ x,x\} ,\{ x,y\} ,\{ y,x\} ,\{ y,y\} \} $
- $\{ (x,y),(y,x)\} $
- None of these
Correct

Incorrect

Hint

$A = \{ x,y\}$
$A \times A = \{ (x,x),(x,y),(y,x),(y,y)\}$
If $\left| A \right| = n$
Then $\left| {A \times A} \right| = {n^2}$ .
Question 9 of 25

9. Question
Let $\left| A \right| = m,\left| B \right| = n,$ then number of elements in $A \times B$ is equal to
- $mn$
- $m + n$
- ${2^{mn}}$
- ${2^{m + n}}$
Correct

Incorrect

Hint

$\left| A \right| = m,\left| B \right| = n$
$\left| {A \times B} \right| = m \times n = mn$ .
Question 10 of 25

10. Question
Let $\left| A \right| = 2,\left| B \right| = 3,$ then $\left| {P(A \times B)} \right|$ is equal to
- 6
- 5
- 32
- 64
Correct

Incorrect

Hint

$\left| A \right| = 2,\left| B \right| = 3$
$\left| {A \times B} \right| = 2 \times 3 = 6$
$\left| {P(A \times B)} \right| = {2^6} = 64$ .
Question 11 of 25

11. Question
If $\left| A \right| = 2$ and $\left| B \right| = 4,$ then $\left| {A \times B} \right|$ is equal to
- 6
- 8
- 4
- none of these
Correct

Incorrect

Hint

$\left| A \right| = 2,\left| B \right| = 4$
$\left| {A \times B} \right| = 2 \times 4 = 8$ .

.
Question 12 of 25

12. Question
If $A$ and $B$ be any sets, then $(A \cup B) – (A \cap B)$ is equal to
- $ \in (A – B) \cup (B – A)$
- $A – B$
- $B – A$
- None of these
Correct

Incorrect

Hint

$x \in (A \cup B) - (A \cap B)$
$\Leftrightarrow x \in (A \cup B)$ and $x \notin (A \cap B)$
$\Leftrightarrow x \in (A \cup B)$ and $(x \notin A$ or $x \notin B)$
$\Leftrightarrow [x \in A/latex] and [mathjax][latex](x \notin A$ or $x \notin B)]$ or $[x \in B/latex] and [mathjax][latex](x \notin A$ or $x \notin B)]$
$\Leftrightarrow (x \in A$ and $x \notin B)$ or $(x \in B$ and $x \notin A)$
$\Leftrightarrow (x \in A$ and $x \notin B)$ or $(x \in B$ and $x \notin A)$
$\Leftrightarrow x \in A - B$ or $x \in B - A$
$\Leftrightarrow x \in (A - B) \cup (B - A)$
.
Question 13 of 25

13. Question
If $A = \{ x,y\} ,$ then $\left| {P(P(A \times A))} \right|$ is equal to
- ${2^2}$
- ${4^2}$
- ${2^{16}}$
- ${2^4}$
Correct

Incorrect

Hint

$A = \{ x,y\}$
$\left| {A \times A} \right| = 2 \times 2 = 4$
$\left| {P(A \times A)} \right| = {2^4} = 16$
$\left| {P(P(A \times A))} \right| = {2^{16}}$ .
Question 14 of 25

14. Question
If $A$ and $B$ be two sets. $A = \{ x:x \in z, – 4 \le x < 4\} $ and $B = \{ x:x$ is a natural number$\} $ then $n(A – B)$ is equal to
- 3
- 4
- 5
- 8
Correct

Incorrect

Hint

$A = \{ x:x \in z, - 4 \le x < 4\}$ and $B = \{ x:x$ is a natural number $\}$
$\therefore A = \{ - 4, - 3, - 2, - 1,0,1,2,3\} ,B = \{ 1,2,3,...\}$
$A - B = \{ 4, - 3, - 2, - 1,0\}$
$\therefore n(A - B) = 5$ .
Question 15 of 25

15. Question
If $A = \{ 1,2,3,4,5,6\} $ and $B = \{ 2,4,6,…\} $ (set of all even numbers) then $\left| {P(A – B)} \right|$ is equal to
- 6
- 8
- 4
- 3
Correct

Incorrect

Hint

$A = \{ 1,2,3,4,5,6\}$ and $B = \{ 2,4,6,...\}$
$\begin{array}{c}A - B = \{ 1,2,3,4,5,6\} - \{ 2,4,6,...\} \\ = \{ 1,2,3\} \end{array}$
$\therefore n(A - B\} = 3$
$\left| {P(A - B)} \right| = {2^3} = 8$ .
Question 16 of 25

16. Question
If $A = \left\{ {2,4,7,9} \right\}$and $B = \left\{ {4,5,6,7} \right\}$ then $A\Delta B$is
- $\left\{ {4,7} \right\}$
- $\left\{ {2,9,4,5,7} \right\}$
- $\left\{ {2,5,6,9} \right\}$
- None of these
Correct

Incorrect

Hint

$A = \left\{ {2,4,7,9} \right\}$ and $B = \left\{ {4,5,6,7} \right\}$
$\begin{array}{l}A\Delta B = \left( {A \cup B} \right) - \left( {A \cap B} \right)\\ = \left\{ {2,4,5,6,7,9} \right\} - \left\{ {4,7} \right\}\\ = \left\{ {2,5,6,9} \right\}\end{array}$
Question 17 of 25

17. Question
If A and B be sets, then ${\left( {A \cup B} \right)^c}$is equal to
- ${A^c} \cap B$
- $A \cap {B^c}$
- ${A^c} \cap {B^c}$
- ${A^c} \cup {B^c}$
Correct

Incorrect

Hint

Let
$\begin{array}{l}x \in {\left( {A \cup B} \right)^c}\\ \Leftrightarrow x \notin \left( {A \cup B} \right)\\ \Leftrightarrow x \notin A,x \notin B\\ \Leftrightarrow x \in {A^c},x \in {B^c}\\ \Leftrightarrow x \in {A^c} \cap {B^c}\\\therefore {\left( {A \cup B} \right)^c} = {A^c} \cap {B^c}\end{array}$
Question 18 of 25

18. Question
If A and B be any sets, then ${\left( {A \cap B} \right)^c}$is equal to
- ${A^c} \cap {B^c}$
- ${A^c} \cup {B^c}$
- ${A^c} \cup B$
- $A \cap {B^c}$
Correct

Incorrect

Hint

A and B be any sets. Let $x \in {\left( {A \cap B} \right)^c}$
$\begin{array}{l} \Leftrightarrow x \notin \left( {A \cap B} \right)\\ \Leftrightarrow x \notin A,x \notin B\\ \Leftrightarrow x \in {A^c},x \in {B^c}\\ \Leftrightarrow x \in {A^c} \cup {B^c}\\\therefore {\left( {A \cap B} \right)^c} = {A^c} \cup {B^c}\end{array}$
Question 19 of 25

19. Question
If A and B two sets, $\left| {A \cup B} \right| = \left| A \right| + \left| B \right|$, then $A \cap B$is
- A
- B
- $\emptyset $
- ${A^c}$
Correct

Incorrect

Hint

$\left| {A \cup B} \right| = \left| A \right| + \left| B \right| - \left| {A \cap B} \right|$
Here, $\left| {A \cup B} \right| = \left| A \right| + \left| B \right|$
Hence,
$\begin{array}{l}\left| {A \cap B} \right| = 0\\\therefore A \cap B = \emptyset \end{array}$
Question 20 of 25

20. Question
If A, B and C be any sets, $\left| {A \cup B \cup C} \right| = \left| A \right| + \left| B \right| + \left| C \right|$, then $\left| {A \cap B} \right|$ equal to
- 0
- 3
- 4
- none of these
Correct

Incorrect

Hint

We know, $\left| {A \cup B \cup C} \right| = \left| A \right| + \left| B \right| + \left| C \right| - \left| {A \cap B} \right| - \left| {B \cap C} \right| - \left| {C \cap A} \right| + \left| {A \cap B \cap C} \right|$
Here,
$\begin{array}{l}\left| {A \cup B \cup C} \right| = \left| A \right| + \left| B \right| + \left| C \right|\\\therefore A \cap B = 0\end{array}$
Question 21 of 25

21. Question
If $A = \left\{ {x:x \in N,2 \le x \prec 9} \right\}$and $B = \left\{ {x:x \in Z, – 4 \le x \le 3} \right\}$, then $\left| {A\Delta B} \right|$is equal to
- 10
- 15
- 13
- 11
Correct

Incorrect

Hint

$\begin{array}{l}A = \left\{ {x:x \in N,2 \le x \prec 9} \right\}\\A = \left\{ {2,3,4,5,6,7,8} \right\}\\B = \left\{ {x:x \in Z, - 4 \le x \le 3} \right\}\\B = \left\{ { - 4, - 3, - 2, - 1,0,1,2,3} \right\}\\A\Delta B = \left( {A \cup B} \right) - \left( {A \cap B} \right)\\ = \left\{ { - 4, - 3, - 2, - 1,0,1,2,3,4,5,6,7,8} \right\} - \left\{ {2,3} \right\}\\ = \left\{ { - 4, - 3, - 2, - 1,0,1,4,5,6,7,8} \right\}\\\left| {A\Delta B} \right| = 11\end{array}$
Question 22 of 25

22. Question
If $A = \left\{ {2,3,4,5} \right\},B = \left\{ {1,6,9,11} \right\}$and $C = \left\{ {7,8,10} \right\}$, then what is the relation between A, B and C?
- Disjoint Sets
- Overlapping Sets
- Both of these
- None of these
Correct

Incorrect

Hint

$A = \left\{ {2,3,4,5} \right\},B = \left\{ {1,6,9,11} \right\},C = \left\{ {7,8,10} \right\}$ , then $A \cap B \cap C = \emptyset$ . Since, $A \cap B \cap C = \emptyset$ , then A, B and C are disjoint sets.
Question 23 of 25

23. Question
N={Set of natural numbers}
Z={Set of integers}
N*={Set of whole numbers}
What will be the relation between N, N* and Z?
- $N = {N^ * } \cup Z$
- $N \subset {N^ * } \subset Z$
- $Z \subset N \subset {N^ * }$
- None of these
Correct

Incorrect

Hint

$\begin{array}{l}N = \left\{ {1,2,3,...} \right\}\\{N^ * } = \left\{ {0,1,2,3,...} \right\}\\Z = \left\{ {0, \pm 1, \pm 2, \pm 3,...} \right\}\\\therefore N \subset {N^ * } \subset Z\end{array}$
Question 24 of 25

24. Question
If $A = \left\{ {x,y} \right\},B = \left\{ {x,a,b} \right\}$, then $\left( {A \times B} \right) \cap \left( {B \times A} \right)$ is equal to
- $\emptyset $
- A
- B
- $\left\{ {\left( {x,x} \right)} \right\}$
Correct

Incorrect

Hint

$\begin{array}{l}A = \left( {x,y} \right),B = \left\{ {x,a,b} \right\}\\A \times B = \left\{ {\left( {x,x} \right),\left( {x,a} \right),\left( {x,b} \right),\left( {y,x} \right),\left( {y,a} \right),\left( {y,b} \right)} \right\}\\B \times A = \left\{ {\left( {x,x} \right),\left( {x,y} \right),\left( {a,x} \right),\left( {a,y} \right),\left( {b,x} \right),\left( {b,y} \right)} \right\}\\\therefore \left( {A \times B} \right) \cap \left( {B \times A} \right) = \left\{ {\left( {x,x} \right)} \right\}\end{array}$
Question 25 of 25

25. Question
If $A = \left\{ {x:\left| {5x – 9} \right| \prec 6,x \in R} \right\}$, then A is equal to
- $C = \left\{ {x:\left| {x – 9} \right| \prec \frac{6}{5},x \in R} \right\}$
- $D = \left\{ {x:0 \prec x \prec 3,x \in R} \right\}$
- $B = \left\{ {x:\frac{3}{5} \prec x \prec 5,x \in R} \right\}$
- $E = \left\{ {x: – 9 \prec 5x \prec 6,x \in R} \right\}$
Correct

Incorrect

Hint

$\begin{gathered} A = \left\{ {x:\left| {5x - 9} \right| \prec 6,x \in R} \right\} \hfill \\ \because \left| {5x - 9} \right| \hfill \\ \Rightarrow - 6 \prec 5x - 9 \prec 6 \hfill \\ \Rightarrow - 6 + 9 \prec 5x \prec 6 + 9 \hfill \\ \Rightarrow 3 \prec 5x \prec 15 \hfill \\ \Rightarrow \frac{3}{5} \prec x \prec 5 \hfill \\ \therefore B = \left\{ {x:\frac{3}{5} \prec x \prec 5,x \in R} \right\} \hfill \\ \end{gathered}$ .

Mathematics Class XI

Practice Questions 3 – Sets – Class XI

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