Subject Experts Solutions for Chapter: Set Theory, Exercise 1: PRACTICE QUESTIONS (CLASSROOM WING)

If $O=\left\{1\right\}$, then find the number of all possible proper subsets of $O.$

If $N=\{a,b,c,...z\},$ then find all the possible subsets of $N.$

If $n\left(A\right)=20, n(A\cap B)=10\text{ and} n(A\cup B)=70,$ then find $n\left(B\right).$

If $X=\{0,1,2,3,4,5,6,7,8,9,10\} \text{and} Y=\{2,4,6,8,10\}$, then find $X-Y.$

The mean of $36$ observations is $22$. If one observation $22$ is deleted, then find the new mean.

The following steps are involved in solving the above problem. Arrange them in sequential order.

(A)$\therefore \text{New arithmetic mean} =\frac{770}{35}=22$

(B) $$\begin{gathered} \text{Arithmetic mean} \\ =\frac{\text{The sum of observations}}{\text{Total number of the observations}} \\ \Rightarrow 22=\frac{\text{The sum of observations}}{36} \end{gathered}$$

(C) $\text{The sum of observations} =36\times 22=792$

(D) Since one observation $22$ is deleted, the new sum $=792-22=770$ and the number of observation is $36-1 i.e., 35.$

If $A=\{x:x\in W, x\le 10\} and B=\{x:x\in N, x\le 10\},$ then find $n(A\cup B)$.

The following steps are involved in solving the above problem. Arrange them in the sequential order.

(A) $\Rightarrow n(A\cup B)=11$

(B) $A\cup B=\{0,1,2,3,4,5,6,7,8,9,10\}\cup \{1,2,3,4,5,6,7,8,9,10\}$

(C) From the given data $A=\{0,1,2,3,4,5,6,7,8,9,10\} \text{and} B=\{1,2,3,4,5,6,7,8,9,10\}$

(D) $\therefore A\cup B=\{0,1,2,3,4,5,6,7,8,9,10\}$

The mean of $2,12,x,15,20\text{ and} 17\text{ is} 16$, then find the value of $x.$

The following steps are involved in solving the above problem. Arrange them in sequential order.

(A) $16=\frac{2+12+x+15+20+17}{6}$

(B) $96=66+x$

(C) $\Rightarrow x=96-66=30$

(D) We have arithmetic mean 

$=\frac{\text{The sum of observations }}{\text{Total number of observations}}$

In a class there are $100$ students. Of them, $60$ students attend music classes, $40$ students attend dance classes, and $20$ students attend both the classes. Find the number of students who attend neither of the classes. 

The following steps are involved in solving the above problem. Arrange them in sequential order.

(A) 

 $$\begin{gathered} n(M\cup D)=n\left(M\right)+n\left(D\right)-n(M\cap D) \\ \Rightarrow n(M\cup D)=60+40-20=100-20=80 \end{gathered}$$

(B) Number of students who attend neither of the classes $=100-80=20$

(C) $n\left(M\right)=60, n\left(D\right)=40\text{ and} n(M\cap D)=20 \left(\text{given}\right)$

(D) Let $n\left(M\right)$ be the number of students who attend music classes $n\left(D\right)$ be the number of students who attend dance classes.