Properties of Intersection of Sets

Q. Consider the following statements :
$1.N\cup (B\cap Z)=(N\cup B)\cap Z$ for any subset $B$ of $R,$ where $N$ is the set of positive integers, $Z$ is the set of integers, $R$ is the set of real numbers.

$2.$ Let $A=\{n\in N:1\le n\le 24,n\text{is a multiple of}3\}.$ There exists no subset $B$ of $N$ such that the number of elements in $A$ is equal to the number of elements in $B.$

Which of the above statements is/are correct?

1. $1$ only
2. $2$ only
3. Both $1$ and $2$
4. Neither $1$ nor \(2\
   )

Q.

Let $\mathrm{A}=\left\{\mathrm{x}\in \mathrm{W},\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{whole}\mathrm{numbers}\mathrm{and}\mathrm{x}<3\right\},\mathrm{B}=\left\{\mathrm{x}\in \mathrm{N},\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{natural}\mathrm{numbers}\mathrm{and}2\le \mathrm{x}<4\right\}$ and $C=\left\{3,4\right\}$, then how many elements will $(A\cup B)\times C$ contain?

1. $6$

2. $8$

3. $10$

4. $12$

Q. Consider the following statements :
$1.N\cup (B\cap Z)=(N\cup B)\cap Z$ for any subset $B$ of $R,$ where $N$ is the set of positive integers, $Z$ is the set of integers, $R$ is the set of real numbers.

$2.$ Let $A=\{n\in N:1\le n\le 24,n\text{is a multiple of}3\}.$ There exists no subset $B$ of $N$ such that the number of elements in $A$ is equal to the number of elements in $B.$

Which of the above statements is/are correct?

1. $1$ only
2. $2$ only
3. Both $1$ and $2$
4. Neither $1$ nor \(2\
   )

Q.

If X and Y are two sets and $X^{\prime }$ denotes the complement of X, then
$X\cap (X\cup Y{)}^{\prime }$ is equal to

1. X

2. Y

3. $\varphi$

4. $X\cap Y$

Q. For any set $A,A\cap \varnothing =\varnothing .$

1. True
2. False

Q. For any set $A,A\cap A=A.$

1. False
2. True

Q. For any two non-empty sets $A$ and $B$, $(A\cup B{)}^C\cap (A^C\cap B)$ is equal to

1. $A^C$
2. $B^C$
3. $\varphi$
4. $A\cup B$

Q. For all the sets A, B and C,
(A – B) ∩ (C – B) =

1. $\left(A\cap C\right)\cap B\text{'}$
2. $\left(A\cap C\right)-B$
3. $\left(A\cap C\right)\cap B$
4. $\left(A-C\right)-B$

Q. Two sets given such that $A=\{a:a\text{is a prime number}<25\}\&$
$B=\{b:b\in N,10\le b\le 18\},$ the select the correct statements.

1. $A\cap B=\{11,13,17\}$
2. $A\cup B=B\cup A$
3. $A\cap B=B\cap A$
4. $A\subset B$

Q. For three sets $A,B\&C,$
$A\cap (B\cup C)=$

1. $(A\cap B)\cup (A\cap C)$
2. $(A\cup B)\cap (A\cup C)$
3. $(A\cup B)\cup (A\cup C)$

Q. $\text{For any two sets}A\&B\text{if}$
$A\subset B,\text{then}A\cap B=$

1. $A$
2. $B$
3. $A\cup B$

Q. For any two non-empty sets $A$ and $B$, $(A\cup B{)}^C\cap (A^C\cap B)$ is equal to

1. $A^C$
2. $B^C$
3. $\varphi$
4. $A\cup B$

Q. For any two sets $A\&B;A\cap B=$

1. $B\cap A$
2. $B$
3. $A\cup B$

Q.

If $A$, $B$ and $C$ are three sets such that $A\cap B=A\cap CandA\cup B=A\cup C$, then

1. $A=C$

2. $B=C$

3. $A\cap B=\varphi$

4. $A=B$

Q. For three sets $A,B\&C,$
$A\cap (B\cup C)=$

1. $(A\cap B)\cup (A\cap C)$
2. $(A\cup B)\cap (A\cup C)$
3. $(A\cup B)\cup (A\cup C)$