Cardinal Properties

Q.

A survey of 500 television viewers produced the following information : 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games ? How many watch exactly one of the three games ?

Q.

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 reas newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspaper,

Find :

(i) the numbers of people who read at least one of the newspapers.

(ii) the numbers of people who read exactly one newspaper.

Q.

An investigator interviewed $100$ students to determine the performance of three drinks : milk, coffee and tea. The investigator reported that $10$ students take all three drink milk, coffee and tea ; $20$ students take milk and coffee; $25$ students take milk and tea ; $20$ students take coffee and tea; $12$ students take milk only; $5$ students take coffee only and $8$ students take tea only. Then the number of students who did not take any of three drinks is

1. $10$

2. $20$

3. $60$

4. $30$

Q.

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find :

(i) how many drink tea and coffee both

(ii) how many drink coffee but not tea.

Q.

In a class of 175 students the following data shows the number of students one or more subjects. Mathematics 100 ; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23 ; Mathematics , Physics and Chemistry 18. How many students have offered Mathematics alone ?

1. 60

2. 22

3. 35

4. 48

Q. In a college of $300$ students, every student reads $5$ newspapers. If every newspaper is read by $60$ students, then the number of newspaper is

1. exactly $25$
2. at least $30$
3. at most $20$
4. exactly $50$

Q.

Out of $64$ students, the number of students taking Mathematics is $45$ and the number of students taking both Mathematics and Biology is $10$. Then, the number of students taking the only Biology is:

1. $18$

2. $19$

3. $20$

4. None of these

Q. Let $U=\{1,2,3,4,5,6\},A=\{2,3\}$ and $B=\{3,4,5\}$. Find $A^{\prime },B^{\prime },A^{\prime }\cap B^{\prime },A\cup B$ and hence show that $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$.

Q.

Let A and B be two sets such that n (A) = 16, n(B) = 14, $n(A\cup B)$= 25. Then, $n(A\cap B)$ is equal to .

1. 5

2. 30

3. 50

4. none of these

Q. In a group of $800$ people, $500$ can speak Hindi and $320$ can speak English. Let $\alpha$ be the number of people who speak both languages. Then

1. $20\le \alpha \le 320$
2. $\alpha \le 320$
3. $\alpha \ge 320$
4. $320\le \alpha \le 500$

Q.

State whether the following statements are true or false :

(i) 1 $ϵ$ {1, 2, 3} (ii) a $\subset$ {b, c, a}

(iii) {a} $\subset$ {a, b, c}

(iv) {a, b} = {a, a, b, b, a}

(v) The set {x : x +8=8} is the null set.

Q.

In a survey i was found that 21 persons liked product $P_1$, 26 liked product $P_2$ and 29 liked product $P_3$. If 14 persons liked products $P_{1}and{P}_2$ ; 12, persons liked prooduct $P_3$ and $P_1$; 14 persons liked products $P_2$ and $P_3$ and 8 liked all the three products. Find how many liked product $P_3$ only.

Q.

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find :

(i) how manycan speak both Hindi and English

(ii) how many can speak Hindi only

(iii) how many can speak English only.

Q. The number of elements in a set is called

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

Q.

If A and B are two sets such that n(A) = 70, n(B) = 60, $n(A\cup B)$= 110, then $n(A\cap B)$ is equal to

1. 240

2. 50

3. 40

4. 20

Q.

In a survey of 100 persons it was found that 28 read magazine A, 30 read magazine B, 42 read magazine C, 8 read magazines A and B, 10 read magazines A and C, 5 read magazines B and C and 3 read all the three magazines. Find :

(i) How many read none of three magazines ?

(ii) How many read magazine C only ?

Q.

Students are made to stand in a rows. There are $8$ students in a row. What is the rule which gives the total number of students given the number of rows?

Q.

Let A and B be two sets having 3 and 6 elements respectively. Write the minimum number of elements that $A\cup B$ can have.

Q.

The sum of three consecutive even integers is $168$, what are the integers?

Q. If $P\left(A\right)=0.6$ and the greatest value of $P(A\cap B)=0.4$, then the greatest value of $P\left(B\right)$ is $\frac{k}{10}$. The value of $k$ is

Q. If $A=\{3,5,7,9,11\},B=\{7,9,11,13\}$
$C=\{11,13,15\}$ and $D=\{15,17\}$;
$\left(i\right)A\cap B$
$\left(ii\right)B\cap C$
$\left(iii\right)A\cap C\cap D$
$\left(iv\right)A\cap C$
$\left(v\right)B\cap D$
$\left(vi\right)A\cap (B\cup C)$
$\left(vii\right)A\cap D$
$\left(viii\right)A\cap (B\cup D)$
$\left(ix\right)(A\cap B)\cap (B\cup C)$
$\left(x\right)(A\cup D)\cap (B\cup C)$

Q. Fill in the blanks to make each of the following a true statement:
$\left(i\right)A\cup A^{\prime }=....\left(ii\right){\varphi }^{\prime }\cap A=....$
$\left(iii\right)A\cap A^{\prime }=....\left(iv\right)U^{\prime }\cap A=....$

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.

If A and B are two sets such that n (A) = 20, n(B) = 25 and $n(A\cup B)$= 40, then write $n(A\cap B)$.

Q.

For all sets A and B, A - $(A\cap B)$ is equal to ___

1. $A^{\prime }\cap B$

2. $A\cap B^{\prime }$

3. $A^{\prime }\cap B^{\prime }$

4. $(A\cup B{)}^{\prime }$

Q. Let $A=\{1,2,3,4,5,6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces:
$i.)$ 5A $ii.)$ 8A $iii.)$ 0A
$iv.)$ 4A $v.)$ 2A $vi.)$ 10A

Q.

Out of $500$ first-year students , $260$ passed in the first semester and $210$ passed in the second semester. If $170$ did not pass in either semester, how many passed in both semesters $?$

Q.

If A and B are two sets such that n(A) = 115, n(B) = 326, $n(A-B)$ = 47, then writen $(A\cup B)$.

Q.

Of the members of three athletic teams in a certain school, 21 are in the basketball team, 26 in hockey team and 29 in the football team. 14 play hockey and basket ball, 15 play hockey and basket ball, 15 play hockey and football, 12 play football and basketball and 8 play all the three games. How many members are there in all ?

Q.

$x,y,z$ are positive real numbers and $x+y+z=\frac{1}{2}$ then

1. $\left(\frac{1}{2}\right)<\left(1-x\right)\left(1-y\right)\left(1-z\right)<\left(\frac{1}{3}\right)$

2. $\left(\frac{1}{2}\right)<\left(1-x\right)\left(1-y\right)\left(1-z\right)<\left(\frac{2}{3}\right)$

3. $\left(\frac{1}{2}\right)<\left(1-x\right)\left(1-y\right)\left(1-z\right)<1$

4. None of these