Cardinal Number

Q. Let $\bigcup _{i=1}^{50}{X}_i=\bigcup _{i=1}^n{Y}_i=T,$ where each $X_i$ contains $10$ elements and each $Y_i$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets $X_i$'s and exactly $6$ of sets $Y_i$'s, then $n$ is equal to

1. $15$
2. $30$
3. $50$
4. $45$

Q. The number of elements in a set is called

1. total number
2. number set
3. roster notation
4. cardinal number

Q.

Let $1\le m<n\le p$. The number of subsets of the set $A=\left\{1,2,3,........p\right\}$ having $m,n$ as the least and the greatest elements respectively, is

1. $2^{n-m-1}-1$

2. $2^{n-m-1}$

3. $2^{n-m}$

4. $2^{p-n+m-1}$

Q. The cardinal number of a set $A=\{2,3,6,7\}$ is $4.$

1. False
2. True

Q. If $A=\{x:x$ is a letter of the word 'RAMANA'$\}$,
$B=\{x:x$ is a letter of the word 'MISSISSIPPI'$\}$,
$C=\{x:x$ is a letter of the word 'NOOKBOOK'$\}$,
Then relation between cardinality of sets $A,B$ and $C$ is

1. $n\left(A\right)=n\left(B\right)=n\left(C\right)$
2. $n\left(A\right)=2,n\left(B\right)=n\left(C\right)$
3. $n\left(A\right)+n\left(C\right)=n\left(B\right)$
4. $n\left(B\right)>n\left(A\right)>n\left(C\right)$

Q. The cardinal number of the set $A=\{1,2,2,3,4,5,5,6,6,7,7,8\}$ is .

1. $8$
2. $7$
3. $12$

Q. The number of elements in the set $\left\{\right(a,b)|2a^2+3b^2=35,a,b\in Z\}$ is

1. $2$
2. $4$
3. $8$
4. $12$

Q. The cardinal number of the set $Y=\{x:x\le 16\&x\in N\}$ is

1. $16$
2. $15$
3. $17$

Q.

Suppose $A_1,A_2,.....................,A_{30}$ are thirty sets each having 5 elements each and $B_1,B_2,................B_n$ are $n$ sets each with 3 elements each. Let $\bigcup _{i=1}^{30}{A}_i=\bigcup _{j=1}^n{B}_j=S$ and each element of $S$ belongs to exactly 10 of $A_i$'s and exactly 9 of the $B_j$'s. Then $n$ is equal to

1. $15$

2. $3$

3. $45$

4. $50$

Q. In a survey it was found that, the number of people who like only What’s app, only Facebook, both What’s app and Facebook and neither of them are $2n,3n,\frac{69}{n},\frac{69}{3n}$
respectively. Then

1. $n=1$
2. $n=23$
3. number of people who like facebook is $69$
4. number of people who like facebook is $72$

Q. If $A=\{x,x\in Z\text{and}\frac{x^2-4x+3}{x^2-8x+15}\le 0\}$, then $n\left(A\right)=$

Q. The Set $K=\{k:0\le k\le 1,k\in R\},$ where $R$ is the set of all real numbers is

1. an infinite set
2. a finite set
3. a singleton set

Q. If $A=\{x:x$ is a letter in the word 'QUARANTINE'$\}$, then the cardinality of $A$ is

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

Q. Suppose $A_1,A_2,....,A_{30}$ are thirty sets, each having 5 elements and $B_1,B_2,.....,B_n$ are n sets, each with 3 elements.
Let ${\bigcup }_{i=1}^{30}{A}_i={\bigcup }_{i=1}^n{B}_j=S$. Each element of S belongs to exactly 10 of the $A_{i}s$ and exactly 9 of the $B_{i}s$. Then n is equal to

1. 15
2. 3
3. 45
4. 50

Q. Let $A_1,A_2,...,A_m$ be non-empty subsets of $\{1,2,3,...,100\}$ satisfying the following conditions:
(1) the numbers $|A_1|,|A_2|,...,|A_m|$ are distinct ;
(2) $A_1,A_2,...,A_m$ are pairwise disjoint.
(Here $|A|$ denotes the number of elements in the set $A$).
Then the maximum possible value of $m$ is

1. $13$
2. $14$
3. $15$
4. $16$

Q.

In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The no. of newspaper is

1. At least 30

2. At most 20

3. Exactly 25

4. None of these

Q. If $A=\{x,x\in N\text{and}16-x^2\ge 0\}$, then cardinality of set $A$ is