Number of Subset of a Given Set

Q. Let $S=\{1,2,3,...,100\}$. The number of non empty subsets $A$ of $S$ such that the product of elements in $A$ is even is:

1. $2^{100}-1$
2. $2^{50}-1$
3. $2^{50}(2^{50}-1)$
4. $2^{50}+1$

Q.

Set $A$ has $m$ elements and set $B$ has $n$ elements. If the total number of subsets of $A$ Is $112$ more than the total number of subsets of $B$ then the value of $m\times n$ Is:

1. $25$

2. $22$

3. $28$

4. $24$

Q.

Let the function $f:R\to R$ be defined by $f\left(x\right)=cosx,\forall x\in R$. Show that $f$ is neither one-one nor onto.

Q.

Let $R$ be a relation in the set of natural numbers defined by $R=\left\{\overline{)\left(1+x,1+x^2\right)}x<5,x\in N\right\}$ which of the following is false?

1. $R=\left\{\left(2,2\right),\left(3,5\right),\left(4,10\right),\left(5,17\right),\left(6,25\right)\right\}$

2. Domain of $R=\left\{2,3,4,5,6\right\}$

3. Range of $R=\left\{2,5,10,17,26\right\}$

4. (B) and (C) are true

Q. Let $A$ and $B$ be two finite sets. The set $A$ has $480$ more subsets than $B.$ If $A\cap B$ has $2$ elements, then the number of elements in $A\cup B$ is

1. $14$
2. $12$
3. $13$
4. $10$

Q. If $A=\{x:-2\le x<2,x\in Z\}$, then the number of proper subsets of $A$ is

1. $15$
2. $16$
3. $31$
4. $14$

Q. If $A=\{x:x$ is a prime number between $2$ and $16\},$ then number of subsets of $A$ with exactly two elements is

1. $5$
2. $10$
3. $15$
4. $20$

Q. The number of subsets of the power set of a singleton set is

1. $1$
2. $16$
3. $2$
4. $4$

Q.

Two finite ets have m and n elements. The number of subsets of the first set is 112 more than that of the second. The values of ma and n are respectively.

1. 4, 7

2. 7, 4

3. 4, 4

4. 7, 7

Q. Let A = {1, 2, 3}. The total number of distinct relations that can be defined over A is

1. 6
2. 8
3. None of these
4. $2^9$

Q. Assertion :The power set of a set $A$ contains $128$ elements then number of elements in set $A$ are $7$. Reason: Number of subsets of a set $A$ having $n$ elements is equal to $2^n$.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. Let $S=\{1,2,3,...,100\}$. The number of non empty subsets $A$ of $S$ such that the product of elements in $A$ is even is:

1. $2^{100}-1$
2. $2^{50}-1$
3. $2^{50}(2^{50}-1)$
4. $2^{50}+1$

Q. If $A=\{x:x$ is a prime number between $2$ and $16\},$ then number of subsets of $A$ with exactly two elements is

1. $5$
2. $10$
3. $15$
4. $20$

Q. If set $A$ has $n$ elements, then the total number of subset of $A$ is

1. $n$
2. $2^n$
3. $2n$
4. $n^2$

Q. If $A=\{x:-2\le x<2,x\in Z\}$, then the number of proper subsets of $A$ is

1. $14$
2. $15$
3. $16$
4. $31$

Q.

If a set A has n elements, then the total number of subsets of A is

1. n

2. 2n

3. n²

4. 2ⁿ

Q. If $A=\{\left\{2\right\},\left\{\varphi \right\},\varphi \}$ then $P\left(A\right)$ is

1. None of these
2. $\{\left\{2\right\},\left\{\varphi \right\},\left\{\left\{\varphi \right\}\right\}\}$
3. $\{\varphi ,\left\{2\right\},\left\{\varphi \right\},\{\varphi ,\left\{2\right\}\}\},\{\left\{\varphi \right\},\left\{2\right\}\},\{\left\{\varphi \right\},\varphi \},\{\left\{\varphi \right\},\left\{2\right\},\left\{\varphi \right\},\left\{\left\{\varphi \right\}\right\}\}$
4. $\{\{2,\varphi \},\varphi ,\left\{\varphi \right\},\left\{2\right\},\{\varphi ,\left\{2\right\}\}\}$

Q. If $A=\{x\in N:{\log }_{2}x+\frac{3}{{\log }_{2}x}<4,x>1\}$ and $B=\{x\in Z:(4-x^2\left)\right(x^2-8x+15)\ge 0\},$ then the number of subsets of the set $(A\cup B)-(A\cap B)$ is

1. $256$
2. $32$
3. $64$
4. $128$

Q.

List the elements of $A\cap B$, if $A=\{$ Even numbers $\},B=\{$ Even prime numbers $\}$.

Q. If $A=\{x,y\}$, then the power set of $A$ is

1. $\{x^x,y^y\}$
2. $\{\varphi ,x,y\}$
3. $\{\varphi ,\left\{x\right\},\left\{2y\right\}\}$
4. $\{\varphi ,\left\{x\right\},\left\{y\right\},\{x,y\}\}$

Q. A set contains 2n+1 elements. The number of subsets of this set containing more than n elements is equal to

1. $2^n$
2. $2^{n-1}$
3. $2^{n+1}$
4. $2^{2n}$

Q. The number of non-empty subsets of the set $\{1,2,3,4\}$ is

1. 17
2. 15
3. 14
4. 16

Q. Let $S=\{0,1,5,4,7\}$. Then the total number of subsets of S is

1. 64
2. 20
3. 40
4. 32

Q.

A set S has 7 elements. How many subsets having at most 5 elements does it have?

1. 128

2. 125

3. 120

4. 116

Q. If $A=\{x\in N:{\log }_{2}x+\frac{3}{{\log }_{2}x}<4,x>1\}$ and $B=\{x\in Z:(4-x^2\left)\right(x^2-8x+15)\ge 0\},$ then the number of subsets of the set $(A\cup B)-(A\cap B)$ is

1. $32$
2. $64$
3. $128$
4. $256$

Q. Two finite sets have $m$ and $n$ elements. The total number of subsets of the first set is $56$ more than the total number of subsets of the second set. The values of $m$ and $n$ are respectively.

1. $4,1$
2. $6,3$
3. $8,5$
4. None of these

Q. Two finite sets have $m$ and $n$ number of elements. The total number of subsets of the first set is $56$ more than the total number of subsets of the second set. Then $m$ and $n$ are

1. $7,6$
2. $6,3$
3. $5,1$
4. $8,7$

Q. Let $A$ and $B$ be two sets containing $2$ elements and $4$ elements respectively. The number of subsets of $A\times B$ having $3$ or more elements is :

1. $256$
2. $220$
3. $219$
4. $211$

Q. Let $A$ and $B$ be two finite sets. The set $A$ has $480$ more subsets than $B.$ If $A\cap B$ has $2$ elements, then the number of elements in $A\cup B$ is

1. $14$
2. $13$
3. $12$
4. $10$

Q. Let $Z$ be the set of integers. If
$A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is :

1. $2^{10}$
2. $2^{12}$
3. $2^{18}$
4. $2^{15}$