Cardinality of Sets

Q.

Write all the factors of the following numbers: $\left(\mathrm{a}\right)24\left(\mathrm{b}\right)15\left(\mathrm{c}\right)21\left(\mathrm{d}\right)27\left(\mathrm{e}\right)12\left(\mathrm{f}\right)20\left(\mathrm{g}\right)18\left(\mathrm{h}\right)23\left(\mathrm{i}\right)36$

Q.

Each set X, contains 5 elements and each set Y, contains 2 elements and ${\bigcup }_{r=2}^{20}{X}_r=S={\bigcup }_{r=1}^n{Y}_r$. If each element of S belongs to exactly 10 of the $X_r^{{}^{\prime }s}$ and to exactly 4 of $Y_r^{{}^{\prime }s}$ then find the value of n.

Q.

Two finite sets have m and n elements. The number of elements in the power set of first set is 48 more than the total number of elements in power set of the second set.Then, the values of m and n are :

1. 7, 6

2. 6, 3

3. 6, 4

4. 7, 4

Q.

Write the set of all positive integers whose cube is odd.

Q. If $A=\{x,x\in Z\text{and}|x^2+5x-6|=6-5x-x^2\}$, then cardinality of set $A$ is

Q.

If A and B are two disjoint sets, then $n(A\cup B)$ is equal to

1. n(A) n(B)

2. n(A)+n(B)

3. $n\left(A\right)+n\left(B\right)-n(A\cap B)$

4. $n\left(A\right)+n\left(B\right)+n(A\cap B)$

Q. If $X$ and $Y$ are two sets such that $X\cup Y$ has $50$ elements, $X$ has $28$ elements and $Y$ has $32$ elements, how many elements does $X\cap Y$ have?

Q.

How many elements has $P\left(A\right)$, if $A=\varnothing$?

Q. The cardinality of set $A=\left\{\right(a,b):ab=36\text{where}a\text{and}b\text{are coprime to each other}\}$ is

Q. If $A=\{1,2,2,1,3,4,3,4\}$, then $n\left(A\right)=$

1. $0$
2. $9$
3. $8$
4. $4$

Q. Find the missing term that should be placed in place of ? for a logical pattern given below.



\begin{tabular}{|c|c|c|} \hline 7 & 22 & 5 \\ \hline 5 & 9 & 11 \\ \hline 9 & 11 & 15 \\ \hline 38 & 77 & \( ? \) \\ \hline \end{tabular}

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

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

Q.

Let A and B be two sets having 4 and 7 elements respectively. Then write the maximum number of elements that $A\cup B$ can have.

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. $\begin{array}{cccc}\text{(i)}& \{1,2,3,6\}& \text{(a)}& \left\{x:x\text{is a prime number and divisor of}6\right\}\\ \text{(ii)}& \{2,3\}& \text{(b)}& \left\{x:x\text{is an odd natural number less than}10\right\}\\ \text{(iii)}& \left\{M,A,T,H,E,I,C,S\right\}& \text{(c)}& \left\{x:x\text{is a natural number and divisor of}6\right\}\\ \text{(iv)}& \left\{\text{1, 3, 5, 7, 9}\right\}& \text{(d)}& \left\{x:x\text{is a letter of the word}MATHEMATICS\right\}\end{array}$

Q. Which of the following statement is incorrect?

1. Set of all real numbers and the set of integers are not equivalent sets
2. Two equal sets will always be equivalent sets
3. All the null sets are equivalent to each other
4. Two equivalent sets will always be equal sets

Q. $A$ is set of letters of the word 'SARIKA', $B$ is set of letters of the word 'RAKASI', then $A$ and $B$ are

1. equal and equivalent
2. neither equal nor equivalent
3. equivalent but not equal
4. equal but not equivalent

Q. If $A=\{x,x\in R\text{and}{x}^2-4x+1=0\}$, then $n\left(A\right)=$

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. If $A=\left\{\right(a,b):a^2+b^2=25\text{and}a,b\in N\}$ then $n\left(A\right)=$

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

Q. If the set A is defined as A = {x : x is a factor of 42 and is a prime number}. Then the number of elements in set A is

1. 4
2. 3
3. 6
4. 5

Q. $A$ is set of letters of the word 'NARAYANA'
$B$ is the set of letters of the word 'ARANYA'.
Then which of the following option is incorrect ?

1. $A$ and $B$ are equivalent sets
2. $A$ and $B$ are equal sets
3. $n\left(A\right)=n\left(B\right)$
4. $n\left(A\right)>n\left(B\right)$

Q. Let $S=\{x\in R:x\ge 0\text{and}2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0\}$. Then $S$

1. contains exactly four elements
2. is an empty set
3. contains exactly one element
4. contains exactly two elements

Q. If $A=\{1,2,2,1,3,4,3,4\}$, then $n\left(A\right)=$

Q. Let $A$ and $B$ be two sets such that $n\left(A\right)=3$ and $n\left(B\right)=2$. If $(x,1),(y,2),(z,1)$ are in $A\times B$, then find $A$ and $B$, where $x,y,z$ are distinct elements.

Q. Find the cardinal number of the following sets:
$A_1=\{-2,-1,1,3,5$}

Q.

Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1)\) are in $A\times B,$ find A and B, where x, y, z are distinct elements.

Q. If $A$ is the set of odd prime numbers and $B=\{x\in Z:-6<\frac{2x-5}{3}\le 7\text{and}-4\le \frac{x-7}{2}<4\},$ then the value of $n(A\cap B)$ is

Q. Let S = {a, b, c}. Find the total number of binary operations on S.