Cardinal Properties

Q. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

Q. In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the two drinks. How many people like both coffee and tea?

Q. A survey shows that 63% of the people watch news whereas 76% watch sports. If $x$% of people watch both sports and news, then

1. $x=63$
2. $39\le x\le 63$
3. $x=39$
4. $0\le x\le 39$

Q.

Show that if $A\subset B$ then $C-B\subset C-A.$

Q. In a group of students 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?

Q.

If A and B are subsets of the universal set U, then show that,

$A\subset B\iff A\cup B=B$

Q.

In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(i) If $x\in A$ and $A\in B$ then $x\in B$

(ii) If $A\subset B$ and $B\in C$ then $A\in C$

(iii) If $A\subset B$ and $B\subset C$ then $A\subset C$

(iv) If $A\subset ̸B$ and $B\subset ̸C$ then $A\subset ̸C$

(v) If $x\in A$ and $A\subset ̸B$ then $x\in B$

(vi) If $A\subset B$ and $x\notin B$ then $x\notin A.$

Q. If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T has 11 elements, how many elements does S ∪ T have?

Q.

Examine whether the following statements are true or false :

(i) {a, b} $\subset ̸$ {b, c, a}

(ii) {a, e} $\subset$ {x: x is a vowel in the English alphabet}

(iii) {1, 2, 3} $\subset$ {1, 3, 5}

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

(v) {a} $\in$ {a, b, c}

(vi) {x : x is an even natural number less than 6} $\subset$ {x : x is a natural number which divides 36}

Q. If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X ∩ Y have?

Q. Let $A=\{a,b\},B=\{a,b,c\}.$ Is $A\subset B$? What is $A\cup B$?

Q.

Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?

(i) {3, 4} $\subset$ A (ii) {3, 4} $\in$ A

(iii) {{3, 4}} $\subset$ A (iv) $1\in A$

(v) $1\subset A$ (vi) {1, 2, 5} $\subset$ A

(vii) {1, 2, 5} $\in$ A (viii) {1, 2, 3} $\subset$ A

(ix) $\Phi \in A$ (x) $\Phi \subset A$

(xi) {$\Phi$} $\subset$ A.

Q.

Write the negation of sets A and B are equal if and only if ($A\subseteq B$ and $B\subseteq A$).

Q. Show that if $A\subset B$, then $C-B\subset C-A$

Q.

Using properties of sets, show that

(i) $A\cup (A\cap B)=A$ (ii) $A\cap (A\cup B)=A.$

Q. If A = {1, 3, 5, 7, 9, 11, 13, 15, 17}, B ={2, 4, ......, 18} and N, the set of natural numbers is the universal set, then $(A^{\prime }\cup [(A\cup B)\cap B^{\prime }\left]\right)$ is

1. $\varphi$
2. N
3. A
4. B

Q.

Let A = {a, b} and B = {a, b, c}. Is $A\subset B$ ? What is $A\cup B$ ?

Q.

Find (A - B) $\cup$ (B - A), if A = {1, 3, 4} and B = {2, 5, 9, 11} .

Q. If X and Y are two sets such that n (X) = 17, n (Y) = 23 and n (X ∪ Y) = 38, find n (X ∩ Y).

Q. If $A=\{1,2,3,4\},B=\{3,4,5,6\},C=\{5,6,7,8\}$ and $D=\{7,8,9,10\}$; find
(i) $A\cup B$

Q.

If X and Y are two sets such that n (X) = 17, n (Y) = 23 and $n(X\cup Y)=38,$ find $n(X\cap Y).$

Q.

Assume that $P\left(A\right)=P\left(B\right).$ Show that $A=B.$

Q. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and X ∩ Y has 10 elements, how many elements does Y have?

Q.

If A = {1, 3, 5, 7, 9, 11, 13, 15, 17}, B ={2, 4, ......, 18} and N, the set of natural numbers is the universal set, then $(A^{\prime }\cup [(A\cup B)\cap B^{\prime }\left]\right)$ is

1. $\varphi$

2. A

3. N

4. B

Q. $(A-B)\cap (C-B)=$

1. $(A-C)\cap B$
2. $B-(A\cap C)$
3. $(A\cap C)-B$
4. $A\cap B\cap C$

Q. Find the union of each of the following pairs of sets:
(i) $X=\{1,3,5\}Y=\{1,2,3\}$
(ii) $A=\{a,e,i,o,u\},B=\{a,b,c\}$
(iii) $A=\{x:x\text{is a natural number and multiple of}3\}$.
$B=\{x:x\text{is a natural number less than}6\}$.
(iv) $A=\{x:x\text{is a natural number and}1<x\le 6\}$

Q.

If A = {1, 3, 5, ..... , 17}, B ={2, 4, 6, ..... , 18} and N, the set of natural numbers is the universal set, then $(A^{\prime }\cup [(A\cup B)\cap B^{\prime }\left]\right)$ is

1. $\varphi$

2. N

3. A

4. B

Q.

If A and B are two sets such that n(A)=27, n(B)=35 and$n(A\cup B)=50$, find $n(A\cap B)$.

Q. A and B are finite sets such that n(A)=17 and n(B)=29. If A is a subset of B, then, $n\left(\right(A\cup B)-(A\cap B\left)\right)$= ___ .

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