Power Set

Q. Let $S=\{1,2,3,4,5,6,9\}$. Then the number of elements in the set $T=\{A\subseteq S:A\ne \varphi$ and the sum of all the elements of $A$ is not a multiple of $3\}$ is

Q. Let $S=\{1,2,3,4,5,6,9\}$. Then the number of elements in the set $T=\{A\subseteq S:A\ne \varphi$ and the sum of all the elements of $A$ is not a multiple of $3\}$ is

Q. Let $f\left(x\right)$ be a function defined from $A$ to $B$ where number of elements in $A$ and $B$ are $3$ and $7$ respectively. Then which among the following is/are correct.

1. Number of many-one functions from $B$ to $A$ is $3^7$
2. Total number of functions that can be defined from $B$ to $A$ is $3^7$
3. Total number of functions that can be defined from $A$ to $B$ is $7^3$
4. Number of one-one functions from $A$ to $B$ is $210$

Q. Power set of a set $X$ is a set of all the subsets of $X.$

1. False
2. True

Q. The relation on the set $A=\{x:1<|x|\le 4,x\in Z\}$ is defined by $R=\left\{\right(x,y):y=|x|\}.$ Then the number of elements in the power set of $R$ is

1. $16$
2. $32$
3. $64$
4. $128$

Q. Let set $R=\{P:B\subseteq P\subseteq A\}.$ If $A=\{1,2,3,4,5\}$ and $B=\{1,2\},$ then the number of elements in set $R$ is

1. $2$
2. $4$
3. $8$
4. $10$

Q.

The number of proper subsets of the set $\{1,2,3\}$ is ___.

1. $8$

2. $7$

3. $6$

4. $5$

Q. The number of elements in power set of $A=\{1,2,3,8,9,10\}$ is

1. $32$
2. $64$
3. $36$
4. $12$