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#### Question 4:

Are the following sets equal?
A = {x : x is a letter in the word reap}:
B = {x : x is a letter in the word paper};
C = {x : x is a letter in the word rope}.

A = {r, e, a, p}
B = {p, a, e, r}
C = {r, o, p, e}
Here, A = B because every element of A is a member of B & every element of B is a member of A.
But every element of C is not a member of A & B.
Also, every element of A and B is not a member of C.
Therefore, we can say that these sets are not equal.

#### Question 5:

From the sets given below, pair the equivalent sets:

Two sets A & B are equivalent if their cardinal numbers are equal, i.e., n(A) = n(B).
n(A) = 3
n(B) = 5
n(C) = 3
n(D) =5
Therefore, equivalent sets are (A and C) and (B and D).

#### Question 6:

Are the following pairs of sets equal? Give reasons.
(i) A = {2, 3}, B = {x : x is a solution of x2 + 5x + 6 = 0};
(ii) A = {x : x is a letter of the word " WOLF"};
B = {x : x is a letter of the word " FOLLOW"}.

(i) A = {2, 3}
B = {$-$2, $-$3}
A is not equal to B because every element of A is not a member of B & every element of B is not a member of A.

(ii) A = {W, O, L, F}
B = {F, O, L, W}
Here, A = B because every element of A is a member of B & every element of B is a member of A.

#### Question 7:

From the sets given below, select equal sets and equivalent sets.
A = {0, a}, B = {1, 2, 3, 4} C = {4, 8, 12}, D = {3, 1, 2, 4},
E = {1, 0}, F = {8, 4, 12} G = {1, 5, 7, 11}, H = {a, b}.

Equal sets:
(a) B and D, because every element of B is a member of D & every element of D is a member of B.
(b) C and F, because every element of C is a member of F & every element of F is a member of C

Equivalent sets:
(a) A, E and H      {$\because$ n(A) = n(E) =n(H) = 2}
(b) B, D and G     {$\because$ n(B) = n(D) =n(G) = 4}
(c) C and F           {$\because$ n(C) = n(F)  = 3}

#### Question 8:

Which of the following sets are equal?
A = {x : xN, x, < 3},
B = {1, 2}
C = {3, 1}
D = {x : xN, x is odd, x < 5},
E = {1, 2, 1, 1} F = {1, 1, 3}.

A = {1, 2}
B = {1, 2}
C = {3, 1}
D = {1, 3}
E = {1, 2, 1, 1} = {1, 2}
F = {1, 1, 3} = {1, 3}
A = B = E and C = D = F

#### Question 9:

Show that the set of letters needed to spell "CATARACT" and the set of letters needed to spell "TRACT" are equal.

Letters required to spell CATARACT are {C, A, T, R}. Let this set be denoted as E.
E = {C, A, T, R}
Letters required to spell TRACT are {T, R, A, C}. Let this set be denoted as F.
F = {T, R, A, C}
The two sets E & F are equal because every element of E is a member of F & every element of F is a member of E.

#### Question 1:

Which of the following statements are true? Give reason to support your answer.
(i) For any two sets A and B either
(ii) Every subset of an infinite set is infinite;
(iii) Every subset of a finite set is finite;
(iv) Every set has a proper subset;
(v) {a, b, a, b, a, b, ...} is an infinite set;
(vi) {a, b, c} and {1, 2, 3} are equivalent sets;
(vii) A set can have infinitely many subsets.

#### Question 2:

State whether the following statements are true or false:
(i)
(ii)
(iii)
(iv)
(v) The set {x ; x + 8 = 8} is the null set.

(i) True
(ii) False
It should be written as .
(iii) False
It should be written as .
(iv) True
(v) False
The element of the set {x ; x + 8 = 8} is {0}. Therefore, it is not an empty or null set.

#### Question 3:

Decide among the following sets, which are subsets of which:

We have:

B = {2, 4, 6}
C = {2, 4, 6, 8,...}
D = {6}
Therefore, we can say that D$\subset$A$\subset$B$\subset$C.

#### Question 4:

(i) The set of all integers is contained in the set of all set of all rational numbers.
(ii) The set of all crows is contained in the set of all birds.
(iii) The set of all rectangle is contained in the set of all squares.
(iv) The set of all real numbers is contained in the set of all complex numbers.
(v) The sets P = {a} and B = {{a}} are equal.
(vi) The sets A = {x : x is a letter of the word "LITTLE"} and,
B = {x : x is a letter of the word "TITLE"} are equal.

(i) True
A rational number is any $\frac{m}{n}$, where m and n are any integers (n$\ne$0). Any integer can be put into that form by setting n = 1. Therefore, the set of all integers is contained in the set of all rational numbers.
(ii) True
All crows are birds. Therefore, the set of all crows is contained in the set of all birds.
(iii) False
Every square can be a rectangle, but every rectangle cannot be a square.
(iv) True
Every real number can be written in the (a + bi) form. Thus, we can say that the
set of all real numbers is contained in the set of all complex numbers.
(v) False
P = {a}
B = {{a}} = {P}
P$\ne${P}
(vi) True
We have:
A =
{x:x is a letter of the word LITTLE} = {L, I, T, E}
B = {x:x is a letter of the word TITLE} = {T, I, L, E}
Sets A & B are equal because every element of A is a member of B & every element of B is a member of A.

#### Question 5:

Which of the following statements are correct?
Write a correct form of each of the incorrect statements.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix) $\left\{x:x+3=3\right\}=\varphi$

Here, (viii) is correct.

The correct forms of each of the incorrect statements are:
(i)
(ii)
(iii)
(iv)
(v)
(vi) {a,b}⊄{a,{b,c}}
(vii)
(ix) $\left\{x:x+3=3\right\}\ne \varphi$

#### Question 6:

Let A = {a, b, {c, d}, e}. Which of the following statements are false and why?
(i)
(ii)
(iii)
(iv) $a\in A$
(v) $a\subset A$
(vi)
(vii)
(viii)
(ix) $\varphi \in A$
(x) $\left\{\varphi \right\}\subset A$

A = {a, b, {c, d}, e}
(i) False
The correct statement would be .
(ii) True
(iii) True
(iv) True
(v) False
The correct statement would be {a}⊂ A or a A.
(vi) True
(vii) False
The correct statement would be .
(viii) False
The correct statement would be {a, b, c} ⊄ A.
(ix) False
A null set is a subset of every set. Therefore, the correct statement would be $\varphi \subset A$.
(x) False
$\mathrm{\varphi }$ is an empty set; in other words, this set has no element. It is denoted by $\mathrm{\varphi }$. Therefore, the correct statement would be $\varphi \subset A$.

#### Question 7:

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:
(i) $1\in A$
(ii)
(iii)
(iv)
(v) $\varphi \in A$
(vi) $\varphi \subset A$

(i) False
If it could be 1$\notin$A , then it would be true .

(ii) False
The correct form would be .

(iii) True

(iv) True

(v) False
A null set is a subset of every set. Therefore, the correct form would be .

(vi) True

#### Question 8:

Let . Which of the following are true?
(i) $\varphi \in A$
(ii) $\left\{\varphi \right\}\in A$
(iii) $\left\{1\right\}\in A$
(iv)
(v) $2\subset A$
(vi)
(vii)
(viii)
(ix) $\left\{\left\{\varphi \right\}\right\}\subset A$

(i) True
(ii) True
(iii) False
The correct form would be $\left\{1\right\}\subset A$.
(iv) True
(v) False
The correct form would be 1$\in$A.
(vi) True
(vii) True
(viii) True
(ix) True

#### Question 9:

Write down all possible subsets of each of the following sets:
(i) {a},
(ii) {0, 1},
(iii) {a, b, c},
(iv) {1, {1}},
(v) $\left\{\varphi \right\}$.

#### Question 10:

Write down all possible proper subsets each of the following sets:
(i) {1, 2},
(ii) {1, 2, 3}
(iii) {1}.

(i) {1}, {2}
(ii) {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}
(iii) No proper subsets are there in this set.

#### Question 11:

What is the total number of proper subsets of a set consisting of n elements?

We know that the total number of subsets of a finite set consisting of n elements is 2n.
Therefore, the total number of proper subsets of a set consisting of n elements is 2n$-$1.

#### Question 12:

If A is any set, prove that: $A\subseteq \varphi ⇔A=\varphi .$

To prove: $A\subseteq \mathrm{\varphi }⇔A=\mathrm{\varphi }$
Proof:
Let:
$A\subseteq \varphi$
If A is a subset of an empty set, then A is the empty set.
$A=\varphi$

Now, let $A=\varphi$.
This means that A is an empty set.
We know that every set is a subset of itself.
$A\subseteq \varphi$
Thus, we have:
$A\subseteq \varphi ⇔A=\varphi$

Prove that:

#### Question 14:

How many elements has ?

#### Question 15:

What universal set (s) would you propose for each of the following:
(i) The set of right triangles.
(ii) The set of isosceles triangles.

(i) The set of all triangles in a plane
(ii) The set of all triangles in a plane

#### Question 16:

If , then prove that $X\subseteq Y.$

Given:

To prove:
$X\subseteq Y$

#### Question 1:

What is the difference between a collection and a set? Give reasons to support your answer?

Well-defined collections are sets.
Example:
The collection of good teachers in a school is not a set, It is a collection.
Thus, we can say that every set is a collection, but every collection is not necessarily a set.
The collection of vowels in English alphabets is a set.

#### Question 2:

(i) A collection of all natural numbers less than 50.
(ii) The collection of good hockey players in India.
(iii) The collection of all girls in your class.
(iv) The collection of most talented writers of India.
(v) The collection of difficult topics in mathematics.
(vi) The collection of all months of a year beginning with the letter J.
(Vii) A collection of novels written by Munshi Prem Chand.
(Viii) The collection of all question in this chapter.
(ix) A collection of most dangerous animals of the world.
(x) The collection of prime integers.

(i) The collection of all natural numbers less than 50 is a set because it is well defined.
(ii) The collection of good hockey players is not a set because the goodness of a hockey player is not defined here. So, it is not a set.
(iii) The collection of all girls in a class is a set, as it is well defined that all girls of the class are being talked about.
(iv) The collection of the most talented writers of India is a set because it is well defined.
(v) The collection of difficult topics in mathematics is not a set because a topic can be easy for one student while difficult for the other student.
(vi) The collection of all months of a year beginning with the letter J is a set given by {January, June, July}
(vii) A collection of novels written by Munshi Prem Chand is a set because one can determine whether the novel is written by Munshi Prem Chand or not.
(Viii) The collection of all question in this chapter is a set because one can easily check whether it is a question of the chapter or not.
(ix) A collection of most dangerous animals of the world is not a set because we cannot decide whether the animal is dangerous or not.
(x) The collection of prime integers is set given by {2, 3, 5........}

#### Question 3:

If A = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blanks spaces:
(i) 4 ...... A
(ii) −4 ...... A
(iii) 12 ...... A
(iv) 9 ...... A
(v) 0 ...... A
(vi) −2 ...... A

(i) 4$\in$A
(ii) −4$\notin$A
(iii) 12$\notin$A
(iv) 9$\in$A
(v) 0$\in$A
(vi) −2$\notin$A

#### Question 1:

If A and B are two sets such that $A\subset B$, then find:
(i) $A\cap B$
(ii) $A\cup B$

From the Venn diagrams given below, we can clearly say that if A and B are two sets such that $A\subset B$, then

(i) Form the given Venn diagram, we can see that  $A\cap B$ = A

(ii) Form the given Venn diagram, we can see that  $A\cup B$ = B

#### Question 2:

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:
(i) $A\cup B$
(ii) $A\cup C$
(iii) $B\cup C$
(iv) $B\cup D$
(v) $A\cup B\cup C$
(vi) $A\cup B\cup D$
(vii) $B\cup C\cup D$
(viii) $A\cap \left(B\cup C\right)$
(ix) $\left(A\cap B\right)\cap \left(B\cap C\right)$
(x) $\left(A\cup D\right)\cap \left(B\cup C\right)$.

Given:
A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}
(i) $A\cup B$ = {1, 2, 3, 4, 5, 6, 7, 8}
(ii) $A\cup C$ = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}
(iii) $B\cup C$ = {4, 5, 6, 7, 8, 9, 10, 11}
(iv) $B\cup D$ = {4, 5, 6, 7, 8, 10, 11, 12, 13, 14}
(v) $A\cup B\cup C$ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
(vi) $A\cup B\cup D$ = {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14}
(vii) $B\cup C\cup D$ = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}
(viii) $A\cap \left(B\cup C\right)$ = {4, 5}
(ix) $\left(A\cap B\right)\cap \left(B\cap C\right)$ = $\mathrm{\varphi }$
(x) $\left(A\cup D\right)\cap \left(B\cup C\right)$ = {4, 5, 10, 11}

#### Question 3:

Let and D = {x : x is a prime natural number}. Find:
(i) $A\cap B$
(ii) $A\cap C$
(iii) $A\cap D$
(iv) $B\cap C$
(v) $B\cap D$
(vi) $C\cap D$

D = {x:x is a prime natural number.} = {2, 3, 5, 7,...}
(i) $A\cap B$ = B
(ii) $A\cap C$ = C
(iii) $A\cap D$ = D
(iv) $B\cap C$ = $\mathrm{\varphi }$
(v) $B\cap D$ = {2}
(vi) $C\cap D$ = D$-${2}

#### Question 4:

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find:
(i) $A-B$
(ii) $A-C$
(iii) $A-D$
(iv) $B-A$
(v) $C-A$
(vi) $D-A$
(vii) $B-C$
(viii) $B-D$

Given:
A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}
(i) $A-B$  = {3, 6, 15, 18, 21}
(ii) $A-C$ = {3, 15, 18, 21}
(iii) $A-D$ = {3, 6, 12, 18, 21}
(iv) $B-A$ = {4, 8, 16, 20}
(v) $C-A$ = {2, 4, 8, 10, 14, 16}
(vi) $D-A$ = {5, 10, 20}
(vii) $B-C$ = {20}
(viii) $B-D$ = {4, 8, 12, 16}

#### Question 5:

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find
(i) $A\text{'}$
(ii) $B\text{'}$
(iii) $\left(A\cap C\right)\text{'}$
(iv) $\left(A\cup B\right)\text{'}$
(v) $\left(A\text{'}\right)\text{'}$
(vi) $\left(B-C\right)\text{'}$

Given:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B= {2, 4, 6, 8} and C = {3, 4, 5, 6}
(i) $A\text{'}$ = {5, 6, 7, 8, 9}
(ii) $B\text{'}$ = {1, 3, 5, 7, 9}
(iii) $\left(A\cap C\right)\text{'}$ = {1, 2, 5, 6, 7, 8, 9}
(iv) $\left(A\cup B\right)\text{'}$ = {5, 7, 9}
(v) $\left(A\text{'}\right)\text{'}$ = {1, 2, 3, 4} = A
(vi) $\left(B-C\right)\text{'}$ = {1, 3, 4, 5, 6, 7, 9}

#### Question 6:

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) $\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$
(ii) $\left(A\cap B\right)\text{'}=A\text{'}\cup B\text{'}$.

Given:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}
We have to verify:

(i) $\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$
LHS

RHS
$A\text{'}=\left\{1,3,5,7,9\right\}\phantom{\rule{0ex}{0ex}}B\text{'}=\left\{1,4,6,8,9\right\}\phantom{\rule{0ex}{0ex}}A\text{'}\cap B\text{'}=\left\{1,9\right\}$

LHS = RHS
Hence proved.

(ii) $\left(A\cap B\right)\text{'}=A\text{'}\cup B\text{'}$
LHS
$A\cap B=\left\{2\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap B\right)\text{'}=\left\{1,3,4,5,6,7,8,9\right\}\phantom{\rule{0ex}{0ex}}$

RHS
$A\text{'}=\left\{1,3,5,7,9\right\}\phantom{\rule{0ex}{0ex}}B\text{'}=\left\{1,4,6,8,9\right\}\phantom{\rule{0ex}{0ex}}A\text{'}\cup B\text{'}=\left\{1,3,4,5,6,7,8,9\right\}$

LHS = RHS
Hence proved.

#### Question 1:

Find the smallest set A such that .

We have to find the smallest set A such that .

The union of the two sets A & B is the set of all those elements that belong to A or to B or to both A & B.

Thus, A must be {3, 5, 9}.

#### Question 2:

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities:
(i) $A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right)$
(ii) $A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$
(iii) $A\cap \left(B-C\right)=\left(A\cap B\right)-\left(A\cap C\right)$
(iv) $A-\left(B\cup C\right)=A\left(A-B\right)\cap \left(A-C\right)$
(v) $A-\left(B\cap C\right)=\left(A-B\right)\cup \left(A-C\right)$
(vi) $A\cap \left(B∆C\right)=\left(A\cap B\right)∆\left(A\cap C\right)$.

Given:
A = {1, 2, 4, 5}, B = {2, 3, 5, 6} and C = {4, 5, 6, 7}
We have to verify the following identities:
(i) $A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right)$

LHS
$\left(B\cap C\right)=\left\{5,6\right\}\phantom{\rule{0ex}{0ex}}A\cup \left(B\cap C\right)=\left\{1,2,4,5,6\right\}\phantom{\rule{0ex}{0ex}}$

RHS
$\left(A\cup B\right)=\left\{1,2,3,4,5,6\right\}\phantom{\rule{0ex}{0ex}}\left(A\cup C\right)=\left\{1,2,4,5,6,7\right\}\phantom{\rule{0ex}{0ex}}\left(A\cup B\right)\cap \left(A\cup C\right)=\left\{1,2,4,5,6\right\}$

LHS = RHS
∴ $A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right)$

(ii) $A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$

LHS
$\left(B\cup C\right)=\left\{2,3,4,5,6,7\right\}\phantom{\rule{0ex}{0ex}}A\cap \left(B\cup C\right)=\left\{2,4,5\right\}$

RHS
$A\cap B=\left\{2,5\right\}\phantom{\rule{0ex}{0ex}}A\cap C=\left\{4,5\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap B\right)\cup \left(A\cap C\right)=\left\{2,4,5\right\}\phantom{\rule{0ex}{0ex}}$

LHS = RHS
$A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$

(iii) $A\cap \left(B-C\right)=\left(A\cap B\right)-\left(A\cap C\right)$

LHS

RHS
$\left(A\cap B\right)=\left\{2,5\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap C\right)=\left\{4,5\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap B\right)-\left(A\cap C\right)=\left\{2\right\}$

LHS = RHS
$A\cap \left(B-C\right)=\left(A\cap B\right)-\left(A\cap C\right)$

(iv) $A-\left(B\cup C\right)=\left(A-B\right)\cap \left(A-C\right)$

LHS
$\left(B\cup C\right)=\left\{2,3,4,5,6,7\right\}\phantom{\rule{0ex}{0ex}}A-\left(B\cup C\right)=\left\{1\right\}\phantom{\rule{0ex}{0ex}}$

RHS
$\left(A-B\right)=\left\{1,4\right\}\phantom{\rule{0ex}{0ex}}\left(A-C\right)=\left\{1,2\right\}\phantom{\rule{0ex}{0ex}}\left(A-B\right)\cap \left(A-C\right)=\left\{1\right\}\phantom{\rule{0ex}{0ex}}$

LHS = RHS
∴ $A-\left(B\cup C\right)=\left(A-B\right)\cap \left(A-C\right)$

(v) $A-\left(B\cap C\right)=\left(A-B\right)\cup \left(A-C\right)$

LHS
$\left(B\cap C\right)=\left\{5,6\right\}\phantom{\rule{0ex}{0ex}}A-\left(B\cap C\right)=\left\{1,2,4\right\}\phantom{\rule{0ex}{0ex}}$

RHS
$\left(A-B\right)=\left\{1,4\right\}\phantom{\rule{0ex}{0ex}}\left(A-C\right)=\left\{1,2\right\}\phantom{\rule{0ex}{0ex}}\left(A-B\right)\cup \left(A-C\right)=\left\{1,2,4\right\}$

LHS = RHS
$A-\left(B\cap C\right)=\left(A-B\right)\cup \left(A-C\right)$

(vi) $A\cap \left(B∆C\right)=\left(A\cap B\right)∆\left(A\cap C\right)$

LHS
$\left(B∆C\right)=\left(B-C\right)\cup \left(C-B\right)\phantom{\rule{0ex}{0ex}}\left(B-C\right)=\left\{2,3\right\}\phantom{\rule{0ex}{0ex}}\left(C-B\right)=\left\{4,7\right\}\phantom{\rule{0ex}{0ex}}\left(B-C\right)\cup \left(C-B\right)=\left\{2,3,4,7\right\}\phantom{\rule{0ex}{0ex}}⇒\left(B∆C\right)=\left\{2,3,4,7\right\}\phantom{\rule{0ex}{0ex}}A\cap \left(B∆C\right)=\left\{2,4\right\}\phantom{\rule{0ex}{0ex}}$

RHS
$\left(A\cap B\right)=\left\{2,5\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap C\right)=\left\{4,5\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap B\right)∆\left(A\cap C\right)=\left\{\left(A\cap B\right)-\left(A\cap C\right)\right\}\cup \left\{\left(A\cap C\right)-\left(A\cap B\right)\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap B\right)-\left(A\cap C\right)=\left\{2\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap C\right)-\left(A\cap B\right)=\left\{4\right\}\phantom{\rule{0ex}{0ex}}\left\{\left(A\cap B\right)-\left(A\cap C\right)\right\}\cup \left\{\left(A\cap C\right)-\left(A\cap B\right)\right\}=\left\{2,4\right\}\phantom{\rule{0ex}{0ex}}⇒\left(A\cap B\right)∆\left(A\cap C\right)=\left\{2,4\right\}\phantom{\rule{0ex}{0ex}}$

LHS = RHS
∴ $A\cap \left(B∆C\right)=\left(A\cap B\right)∆\left(A\cap C\right)$

#### Question 3:

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
(i) $\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$
(ii) $\left(A\cap B\right)\text{'}=A\text{'}B\text{'}.$

Given:
U = {2, 3, 5, 7, 9}
A = {3, 7}
B = {2, 5, 7, 9}

To prove :
(i) $\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$
(ii) $\left(A\cap B\right)\text{'}=A\text{'}\cup B\text{'}$

Proof :

(i) LHS:
$\left(A\cup B\right)=\left\{2,3,5,7,9\right\}\phantom{\rule{0ex}{0ex}}\left(A\cup B\right)\text{'}=\mathrm{\varphi }\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

RHS:

$\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$

(ii) LHS:
$\left(A\cap B\right)=\left\{7\right\}\phantom{\rule{0ex}{0ex}}\left(A\cap B\right)\text{'}=\left\{2,3,5,9\right\}\phantom{\rule{0ex}{0ex}}$

RHS:
$A\text{'}=\left\{2,5,9\right\}\phantom{\rule{0ex}{0ex}}B\text{'}=\left\{3\right\}\phantom{\rule{0ex}{0ex}}A\text{'}\cup B\text{'}=\left\{2,3,5,9\right\}$

LHS = RHS

$\left(A\cap B\right)\text{'}=A\text{'}\cup B\text{'}$

#### Question 4:

For any two sets A and B, prove that

(i) B ⊂ A ∪ B                          (ii) A ∩ A                          (iii) AA ∩ B = A

(i) For all xB

xA or xB

xA ∪ B            (Definition of union of sets)

B ⊂ A ∪ B

(ii) For all A ∩ B

xA and x ∈ B              (Definition of intersection of sets)

xA

⇒ A ∩ A

(iii) Let AB. We need to prove A ∩ B = A.

For all xA

xA and x ∈ B          (AB)

xA ∩ B

AA ∩ B

Also, AA

Thus, AA ∩ B and AA

A ∩ B = A         [Proved in (ii)]

∴ AA ∩ B = A

#### Question 5:

For any two sets A and B, show that the following statements are equivalent:
(i) $A\subset B$
(ii) $A-B=\varphi$
(iii) $A\cup B=B$
(iv) $A\cap B=A.$

We have that the following statements are equivalent:
(i) $A\subset B$
(ii) $A-B=\varphi$
(iii) $A\cup B=B$
(iv) $A\cap B=A$

Proof:

#### Question 6:

For three sets A, B and C, show that
(i) $A\cap B=A\cap C$ need not imply B = C.
(ii) $A\subset B⇒C-B\subset C-A$

(i) Let A = {2, 4, 5, 6},  B = {6, 7, 8, 9} and C = {6, 10, 11, 12,13}

(ii)

#### Question 7:

For any two sets, prove that:
(i) $A\cup \left(A\cap B\right)=A$
(ii) $A\cap \left(A\cup B\right)=A$

(i)

(ii)

#### Question 8:

Find sets A, B and C such that are non-empty sets and $A\cap B\cap C=\varphi$.

Let us consider the following sets,

A = {5, 6, 10 }
B = {6,8,9}
C = {9,10,11}

#### Question 9:

For any two sets A and B, prove that: $A\cap B=\varphi ⇒A\subseteq B\text{'}$.

Let .

$⇒a\in B\text{'}$

Thus, .

#### Question 10:

If A and B are sets, then prove that are pair wise disjoint.

#### Question 11:

Using properties of sets, show that for any two sets A and B,
$\left(A\cup B\right)\cap \left(A\cap B\text{'}\right)=A$

#### Question 12:

For any two sets of A and B, prove that:
(i) $A\text{'}\cup B=U⇒A\subset B$
(ii) $B\text{'}\subset A\text{'}⇒A\subset B$

#### Question 13:

Is it true that for any sets A and ? Justify your answer.

#### Question 14:

Show that for any sets A and B,
(i) $A=\left(A\cap B\right)\cup \left(A-B\right)$
(ii) $A\cup \left(B-A\right)=A\cup B$

(i)

$\mathrm{RHS}=\left(A\cap B\right)\cup \left(A-B\right)\phantom{\rule{0ex}{0ex}}⇒\mathrm{RHS}=\left(A\cap B\right)\cup \left(A\cap B\text{'}\right)\phantom{\rule{0ex}{0ex}}⇒\mathrm{RHS}=\left[\left(A\cap B\right)\cup A\right]\cap \left(\left(A\cap B\right)\cup B\text{'}\right)\phantom{\rule{0ex}{0ex}}⇒\mathrm{RHS}=A\cap \left[\left(A\cup B\text{'}\right)\cap \left(B\cup B\text{'}\right)\right]\phantom{\rule{0ex}{0ex}}⇒\mathrm{RHS}=A\cap \left[\left(A\cup B\text{'}\right)\cap U\right]\phantom{\rule{0ex}{0ex}}⇒\mathrm{RHS}=A\cap \left(A\cup B\text{'}\right)\phantom{\rule{0ex}{0ex}}⇒\mathrm{RHS}=A=\mathrm{LHS}$

(ii)
$\mathrm{LHS}=A\cup \left(B-A\right)\phantom{\rule{0ex}{0ex}}⇒\mathrm{LHS}=A\cup \left(B\cap A\text{'}\right)\phantom{\rule{0ex}{0ex}}⇒\mathrm{LHS}=\left(A\cup B\right)\cap \left(A\cup A\text{'}\right)\phantom{\rule{0ex}{0ex}}⇒\mathrm{LHS}=\left(A\cup B\right)\cap \mathrm{U}\phantom{\rule{0ex}{0ex}}⇒\mathrm{LHS}=\mathrm{A}\cup \mathrm{B}=\mathrm{RHS}$

#### Question 15:

Each set X, contains 5 elements and each set Y, contains 2 elements and $\underset{r=1}{\overset{20}{\cup }}{X}_{r}=S=\underset{r=1}{\overset{n}{\cup }}{Y}_{r}$. If each element of S belong to exactly 10 of the Xr's and to eactly 4 of Yr's, then find the value of n.

It is given that each set X contains 5 elements and $\underset{r=1}{\overset{20}{\cup }}{X}_{r}=S$.

$\therefore n\left(S\right)=20×5=100$

But, it is given that each element of S belong to exactly 10 of the Xr's.

∴ Number of distinct elements in S = $\frac{100}{10}=10$           .....(1)

It is also given that each set Y contains 2 elements and $\underset{r=1}{\overset{n}{\cup }}{Y}_{r}=S$.

$\therefore n\left(S\right)=n×2=2n$

Also, each element of S belong to eactly 4 of Yr's.

∴ Number of distinct elements in S = $\frac{2n}{4}$                  .....(2)

From (1) and (2), we have

$\frac{2n}{4}=10\phantom{\rule{0ex}{0ex}}⇒n=20$

Hence, the value of n is 20.

#### Question 1:

For any two sets A and B, prove that : $A\text{'}-B\text{'}=B-A$

So, LHS = RHS

#### Question 2:

For any two sets A and B, prove the following:
(i) $A\cap \left(A\text{'}\cup B\right)=A\cap B$
(ii) $A-\left(A-B\right)=A\cap B$
(iii) $A\cap \left(A\cup B\right)\text{'}=\varphi$
(iv) .

(i)

Hence proved.

(ii)

Hence proved.

(iii)

Hence proved.

(iv)
.

Hence proved.

#### Question 3:

If A, B, C are three sets such that $A\subset B$, then prove that
$C-B\subset C-A$.

#### Question 4:

For any two sets A and B, prove that

(i) $\left(A\cup B\right)-B=A-B$
(ii) $A-\left(A\cap B\right)=A-B$
(iii) $A-\left(A-B\right)=A\cap B$
(iv) $A\cup \left(B-A\right)=A\cup B$                                  [NCERT EXEMPLAR]
(v) $\left(A-B\right)\cup \left(A\cap B\right)=A$                                 [NCERT EXEMPLAR]

(i)

(ii)

(iii)

(iv)

(v)

#### Question 1:

If A and B are two sets such that , find .

We know:
$n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)\phantom{\rule{0ex}{0ex}}⇒50=28+32-n\left(A\cap B\right)\phantom{\rule{0ex}{0ex}}⇒n\left(A\cap B\right)=60-50=10$

#### Question 2:

If P and Q are two sets such that P has 40 elements, $P\cup Q$ has 60 elements and $P\cap Q$ has 10 elements, how many elements does Q have?

#### Question 3:

In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach physics and mathematics. How many teach physics?

Let A be the number of teachers who teach mathematics & B be the number of teachers who teach physics.

#### Question 4:

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 like both coffee and tea?

Let A denote the set of the people who like tea & B denote the set of the people who like coffee.

#### Question 5:

Let A and B be two sets such that : . Find
(i) $n\left(B\right)$
(ii)
(iii)

Given:

#### Question 6:

A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?

Let A & B denote the sets of the Indians who like oranges & bananas, respectively.

#### Question 7:

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
(i) how many can speak both Hindi and English:
(ii) how many can speak Hindi only;
(iii) how many can speak English only.

Let A & B denote the sets of the persons who like Hindi & English, respectively.

#### Question 8:

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find:
(i) how may drink tea and coffee both;
(ii) how many drink coffee but not tea.

Let A & B denote the sets of the persons who drink tea & coffee, respectively .

#### Question 9:

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the numbers of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.

#### Question 10:

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 football, 12 play football and basketball and 8 play all the three games. How many members are there in all?

Let A, B & C be the sets of members in basketball team, hockey team & football team, respectively.

Therefore, there are 43 members in all teams.

#### Question 11:

In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?

Let A & B denote the sets of the persons who can speak Hindi & Bengali, respectively.

#### Question 12:

$\cap$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?

Let F, H B denote the sets of students who watch football, hockey and basketball, respectively.
Also, let U be the universal set.
We have:
n(F) = 285, n(H) = 195, n(B) = 115, n(F$\cap$B) = 45, n(F$\cap$H) = 70 and n(H$\cap$B) = 50
Also, we know:
n(F$\text{'}$$\cap$H$\text{'}$$\cap$B$\text{'}$) = 50
$⇒$n(F$\cup$H$\cup$B)'= 50
$⇒$n(U) $-$ n(F$\cup$H$\cup$B) = 50
$⇒$500 $-$ n(F$\cup$H$\cup$B) = 50
$⇒$n(F$\cup$H$\cup$B) = 450

Number of students who watch all three games = n(F$\cap$H$\cap$B)
$⇒$n(F$\cup$H$\cup$B) $-$ n(F) $-$ n(H) $-$ n(B) + n(F$\cap$B) + n(F$\cap$H) + n(H$\cap$B)
$⇒$450 $-$ 285 $-$ 195 $-$ 115 + 45 + 70 + 50
$⇒$20

Number of students who watch exactly one of the three games
= n(F) + n(H) + n(B) $-$ 2{n(F$\cap$B) + n(F$\cap$H) + n(H$\cap$B)} + 3{n(F$\cap$H$\cap$B)}
= 285 + 195 + 115 $-$ 2(45 + 70 + 50) + 3(20)
= 325

#### Question 13:

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?

Let A, B C be the sets of the persons who read magazines A, B and C, respectively. Also, let U denote the universal set.
We have: n(U)  = 100
n(A) = 28, n(B) = 30, n(C) = 42, n(A$\cap$B) = 8, n(A$\cap$C) = 10, n(B$\cap$C) = 5 and n(A$\cap$B$\cap$C) = 3
Now,
Number of persons who read none of the three magazines = n(A$\text{'}$$\cap$B$\text{'}$$\cap$C$\text{'}$)
= n(A$\cup$B$\cup$C)$\text{'}$
= n(U) $-$ n(A$\cup$B$\cup$C)
= n(U) $-$ {n(A) + n(B) + n(C) $-$ n(A$\cap$B) $-$ n(A$\cap$C) $-$ n(B$\cap$C) + n(A$\cap$B$\cap$C)}
= 100 $-$ (28 + 30 + 42 $-$$-$ 10 $-$ 5 + 3)
= 20

Number of students who read magazine C only = n(C$\cap$A$\text{'}$$\cap$B$\text{'}$)
= n{C$\cap$(A$\cup$B)$\text{'}$}
= n(C) $-$ n{C$\cap$(A$\cup$B)}
= n(C) $-$ n{(C$\cap$A) $\cup$ (C$\cap$B)}
n(C) $-$ n{(C$\cap$A) + (C$\cap$B)$-$(A$\cap$B$\cap$C)}
= 42 $-$ (10 + 5 $-$ 3)
= 30

#### Question 14:

In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:
(i) How many students were studying Hindi?
(ii) How many students were studying English and Hindi?

Let E, H and S be the sets of students who study English, Hindi and Sanskrit, respectively.
Also, let U be the universal set.
Now, we have:
n(E) = 26, n(S) = 48, n(E$\cap$S) = 8 and n(S$\cap$H) = 8
Also,
n(E$\cap$H$\text{'}$) = 23
$⇒$n(E) $-$ n(E$\cap$H) = 23
$⇒$26 $-$ n(E$\cap$H) = 23
$⇒$n(E$\cap$H) = 3
Therefore, the number of students studying English and Hindi is 3

n(E$\cap$H$\text{'}$$\cap$S$\text{'}$) = 18
$⇒$n(E) $-$ n{E$\cap$(H$\cup$S)'} = 18
$⇒$26 $-$ n{(E$\cap$H)$\cup$(E$\cap$S)}= 18
$⇒$26 $-$ {3 + 8 $-$ n(E$\cap$H$\cap$S)} = 18
$⇒$n(E$\cap$H$\cap$S) = 3

Also,
n(E$\text{'}$$\cap$H$\text{'}$$\cap$S$\text{'}$) = 24
$⇒$n(U) $-$ n(E$\cup$H$\cup$S) = 24
$⇒$n(E$\cup$H$\cup$S) = 76

∴ Number of students studying Hindi = n(E$\cup$H$\cup$S) $-$ n(E) $-$ n(S) + n(E$\cap$H) + n(E$\cap$S) + n(S$\cap$H) $-$ n(E$\cap$H$\cap$S)
= 76 $-$ 24 $-$ 48 + 3 + 8 + 8 $-$ 3
= 18

#### Question 15:

In a survey it was found that 21 persons liked product P1, 26 liked product P2 and 29 liked product P3. If 14 persons liked products P1 and P2; 12 persons liked product P3 and P1 ; 14 persons liked products P2 and P3 and 8 liked all the three products. Find how many liked product P3 only.

Let denote the sets of persons liking products , respectively.
Also, let U be the universal set.
Thus, we have:
n(${P}_{1}$) = 21, n(${P}_{2}$) = 26 and n(${P}_{3}$) = 29
And,
n(${P}_{1}$$\cap$${P}_{2}$) = 14, n(${P}_{1}$$\cap$${P}_{3}$) = 12, n(${P}_{2}\cap {P}_{3}$) = 14 and n(${P}_{1}\cap {P}_{2}\cap {P}_{3}$) = 8

Now,
Number of people who like only product ${P}_{3}$:

Therefore, the number of people who like only product ${P}_{3}$ is 11

#### Question 1:

If a set contains n elements, then write the number of elements in its power set.

A set having n elements has ${2}^{n}$ subsets or elements.

#### Question 2:

Write the number of elements in the power set of null set.

We know that a set of n elements has ${2}^{n}$ subsets or elements.
A null set has no element(s) in it.
∴ Number of elements in the power set of null set =

#### Question 3:

Let A = {x : xN, x is a multiple of 3} and B = {x : xN and x is a multiple of 5}. Write $A\cap B$.

A = {x:xN and x is a multiple of 3.}
= {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45,...}
B = {x:xN and x is a multiple of 5.}
={5, 10, 15, 20, 25, 30, 35, 40, 45,...}
Thus, we have:
$A\cap B$ = {15, 30, 45,...}
= {x:xN, where x is a multiple of 15.}

#### Question 4:

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.

#### Question 5:

If A = {xC : x2 = 1} and B = {xC : x4 = 1}, then write AB and BA.

We have:
A = {xC : x2 = 1}
$⇒$A = {$-$1, 1}
And,
B = {xC : x4 = 1}
$⇒$B = {}
$⇒$B = {=0}
$⇒$B = {$-$1, 1, $-$i, i}
Thus, we get:
AB =$\varnothing$
And,
BA = {$-$i, i}

#### Question 6:

If A and B are two sets such that $A\subset B$, then write B' − A' in terms of A and B.

#### Question 7:

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.

#### Question 8:

If and , then write $A\cap B$.

We have:

= {, ...}
And,

=
Thus, we get:
$A\cap B$ = $\varnothing$

#### Question 9:

If , then write $A\cap B$.

We have:
A =
B =
Thus, we get:
A$\cap$B =

#### Question 10:

If A and B are two sets such that and , then write .

We have:

We know:

= 20 + 25 $-$ 40
= 5

#### Question 11:

If A and B are two sets such that then write .

Thus, we get:

= 115 + 326 $-$ 68
= 373

#### Question 1:

For any set A, (A')' is equal to
(a) A'
(b) A
(c) ϕ
(d) none of these.

(b) A

The complement of the complement of a set is the set itself.

#### Question 2:

Let A and B be two sets in the same universal set. Then, $A-B=$
(a) $A\cap B$
(b) $A\text{'}\cap B$
(c) $A\cap B\text{'}$
(d) none of these.

(c) $A\cap B\text{'}$
A$-$B belongs to those elements of A that do not belong to B.
A$-$B = $A\cap B\text{'}$

#### Question 3:

The number of subsets of a set containing n elements is
(a) n
(b) 2n − 1
(c) n2
(d) 2n

(d) 2n

The total number of subsets of a finite set consisting of n elements is 2n.

#### Question 4:

For any two sets A and B, $A\cap \left(A\cup B\right)=$
(a) A
(b) B
(c) ϕ
(d) none of these.

(a) A

#### Question 5:

If A = {1, 3, 5, B} and B = {2, 4}, then
(a)
(b)
(c)
(d) none of these.

(d) none of these

$4\notin A\phantom{\rule{0ex}{0ex}}$
{4} ⊄ A
B A
Thus, we can say that none of these options satisfy the given relation.

#### Question 6:

The symmetric difference of A and B is
(a) $\left(A-B\right)\cap \left(B-A\right)$
(b) $\left(A-B\right)\cup \left(B-A\right)$
(c) $\left(A\cup B\right)-\left(A\cap B\right)$
(d) $\left\{\left(A\cup B\right)-A\right\}\cup \left\{\left(A\cup B\right)-B\right\}$

(b) $\left(A-B\right)\cup \left(B-A\right)$
The symmetric difference of A  and B is given by :-
$\left(A-B\right)\cup \left(B-A\right)$

#### Question 7:

The symmetric difference of A = {1, 2, 3} and B = {3, 4, 5} is
(a) {1, 2}
(b) {1, 2, 4, 5}
(c) {4, 3}
(d) {2, 5, 1, 4, 3}

(b) {1, 2, 4, 5}
Here,
A = {1, 2, 3} and B = {3, 4, 5}
The symmetric difference of A  and B is given by :-
$\left(A-B\right)\cup \left(B-A\right)$
Now, we have:
$\left(A-B\right)=\left\{1,2\right\}\phantom{\rule{0ex}{0ex}}\left(B-A\right)=\left\{4,5\right\}\phantom{\rule{0ex}{0ex}}\left(A-B\right)\cup \left(B-A\right)=\left\{1,2,4,5\right\}$

#### Question 8:

For any two sets A and B, $\left(A-B\right)\cup \left(B-A\right)=$
(a) $\left(A-B\right)\cup A$
(b) $\left(B-A\right)\cup B$
(c) $\left(A\cup B\right)-\left(A\cap B\right)$
(d) $\left(A\cup B\right)\cap \left(A\cap B\right)$.

(c) $\left(A\cup B\right)-\left(A\cap B\right)$

#### Question 9:

Which of the following statements is false:
(a) $A-B=A\cap B\text{'}$
(b) $A-B=A-\left(A\cap B\right)$
(c) $A-B=A-B\text{'}$
(d) $A-B=\left(A\cup B\right)-B.$

(c) $A-B=A-B\text{'}$
It includes all those elements of A which do not belongs to complement of B which is equal to A$\cap$B but not equal to
A$-$B .
Therefore, (c) is false .

#### Question 10:

For any three sets A, B and C
(a) $A\cap \left(B-C\right)=\left(A\cap B\right)-\left(A\cap C\right)$
(b) $A\cap \left(B-C\right)=\left(A\cap B\right)-C$
(c) $A\cup \left(B-C\right)=\left(A\cup B\right)\cap \left(A\cup C\text{'}\right)$
(d) $A\cup \left(B-C\right)=\left(A\cup B\right)-\left(A\cup C\right).$

(a) $A\cap \left(B-C\right)=\left(A\cap B\right)-\left(A\cap C\right)$

Let x be any arbitrary element of A$\cap \left(B-C\right)$.
Thus, we have,
x$\in \left[A\cap \left(B-C\right)\right]$$⇒$ x $\in A$ and x $\in \left(B-C\right)$

#### Question 11:

Let . Then,
(a) (4, 5]
(b) (4, 5)
(c) [4, 5)
(d) [4, 5]

(c) [4, 5)

$A\cap B=\left[4,5\right)$

#### Question 12:

Let U be the universal set containing 700 elements. If A, B are sub-sets of U such that . Then
(a) 400
(b) 600
(c) 300
(d) none of these.

(c) 300
n($A\text{'}\cap B\text{'}$) = $n\left(A\cup B\right)\text{'}$

#### Question 13:

Let A and B be two sets that . Then, is equal to
(a) 30
(b) 50
(c) 5
(d) none of these

We know:

Now,

= 16 + 14 $-$ 25
= 5

#### Question 14:

If A = |1, 2, 3, 4, 5|, then the number of proper subsets of A is
(a) 120
(b) 30
(c) 31
(d) 32

(c) 31
The number of proper subsets of any set is given by the formula ${2}^{n}-1$, where n is the number of elements in the set.
Here,
n = 5
∴ Number of proper subsets of A =

#### Question 15:

In set-builder method the null set is represented by
(a) { }
(b) Φ
(c)
(d)

(c) $\left\{x:x\ne x\right\}$

#### Question 16:

$\cap$If A and B are two disjoint sets, then is equal to
(a)
(b)
(c)
(d)
(e)

(a)

Two sets are disjoint if they do not have a common element in them, i.e., A$\cap$B = $\varnothing$.

#### Question 17:

For two sets $A\cup B=A$ iff
(a) $B\subseteq A$
(b) $A\subseteq B$
(c) $A\ne B$
(d) $A=B$

(a) $B\subseteq A$
The union of two sets is a set of all those elements that belong to A or to B or to both A and B.
If A$\cup$B = A, then B$\subseteq$A.

#### Question 18:

If A and B are two sets such that , then is equal to
(a) 240
(b) 50
(c) 40
(d) 20

(d) 20
We have:

#### Question 19:

If A and B are two given sets, then $A\cap {\left(A\cap B\right)}^{c}$ is equal to
(a) A
(b) B
(c) Φ
(d) $A\cap {B}^{c}$

(d) $A\cap {B}^{c}$
A and B are two sets.
A$\cap$B is the common region in both the sets.
${\left(\mathrm{A}\cap \mathrm{B}\right)}^{c}$ is all the region in the universal set except A$\cap$B.
Now,
$\mathrm{A}\cap {\left(\mathrm{A}\cap \mathrm{B}\right)}^{c}$ = $\mathrm{A}\cap {\mathrm{B}}^{c}$

#### Question 20:

If A = {x : x is a multiple of 3} and , B = {x : x is a multiple of 5}, then AB is
(a) $A\cap B$
(b) $A\cap \overline{)B}$
(c) $\overline{)A}\cap \overline{)B}$
(d) $\overline{)A\cap B}$

(b) $A\cap \overline{)B}$

A = {x:x is a multiple of 3}
A =

B = {x:x is a multiple of 5.}
B =

Now, we have:
A$-$B =
= $A\cap \overline{)B}$

#### Question 21:

In a city 20% of the population travels by car, 50% travels by bus and 10% travels by both car and bus. Then, persons travelling by car or bus is
(a) 80%
(b) 40%
(c) 60%
(d) 70%

(c) 60%
Suppose C and B represents the population travel by car and Bus respectively.

#### Question 22:

If $A\cap B-B$, then
(a) $A\subset B$
(b) $B\subset A$
(c) $A=\Phi$
(d) $B=\Phi$

(b) $B\subset A$

Only this case is possible.

#### Question 23:

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 drinks milk, coffee and tea; 20 students take milk and coffee; 25 students take milk 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
(a) 10
(b) 20
(c) 25
(d) 30

Disclaimer: The question in the book has some error, so, none of the options are matching with the solution.
The required information is not available in the question.

#### Question 24:

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:
(a) 7, 6
(b) 6, 3
(c) 7, 4
(d) 3, 7

(c) 6, 4

ATQ :

#### Question 25:

In a class of 175 students the following data shows the number of students opting 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?
(a) 35
(b) 48
(c) 60
(d) 22
(e) 30

(c) 60
Let M, P and C denote the sets of students who have opted for mathematics, physics, and chemistry, respectively.
Here,
$n\left(M\right)$ = 100, $n\left(P\right)$ = 70 and $n\left(C\right)$ = 40
Now,

Number of students who opted for only mathematics:

Therefore, the number of students who opted for mathematics alone is 60

#### Question 26:

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 $\underset{i=1}{\overset{30}{\cup }}{A}_{i}=\underset{j=1}{\overset{n}{\cup }}{B}_{j}=S$ and each element of S belong to exactly 10 of the ${A}_{i}\text{'}s$ and exactly 9 of the ${B}_{j}\text{'}s$, then n is equal to

(a) 15                                (b) 3                                 (c) 45                                (d) 35

It is given that each set Ai $\left(1\le i\le 30\right)$ contains 5 elements and $\underset{i=1}{\overset{30}{\cup }}{A}_{i}=S$.

$\therefore n\left(S\right)=30×5=150$

But, it is given that each element of S belong to exactly 10 of the Ai's.

∴ Number of distinct elements in S = $\frac{150}{10}=15$           .....(1)

It is also given that each set Bj $\left(1\le j\le n\right)$ contains 3 elements and $\underset{j=1}{\overset{n}{\cup }}{B}_{j}=S$.

$\therefore n\left(S\right)=n×3=3n$

Also, each element of S belong to eactly 9 of Bj's.

∴ Number of distinct elements in S = $\frac{3n}{9}$                  .....(2)

From (1) and (2), we have

$\frac{3n}{9}=15\phantom{\rule{0ex}{0ex}}⇒n=45$

Thus, the value of n is 45.

Hence, the correct answer is option (c).

#### Question 27:

Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second. The values of m and n are respectively

(a) 4, 7                                  (b) 7, 4                                  (c) 4, 4                                  (d) 7, 7

We know that if a set X contains k elements, then the number of subsets of X are 2k.

It is given that the number of subsets of a set containing m elements is 112 more than the number of subsets of set containing n elements.

Thus, the values of m and n are 7 and 4, respectively.

Hence, the correct answer is option (b).

#### Question 28:

For any two sets A and B, $A\cap \left(A\cup B\right)\text{'}$ is equal to

(a) A                              (b) B                              (c) $\varphi$                              (d) $A\cap B$

Hence, the correct answer is option (c).

#### Question 29:

The set $\left(A\cup B\text{'}\right)\text{'}\cup \left(B\cap C\right)$ is equal to

(a) $A\text{'}\cup B\cup C$                 (b) $A\text{'}\cup B$                    (c) $A\text{'}\cup C\text{'}$                 (d) $A\text{'}\cap B$

Disclaimer: The question seems to be incorrect or there is some printing mistake in the question. The options given in the question does not match with the answer.

#### Question 30:

Let F1 be the set of all parallelograms, F2 the set of all rectangles, F3 the set of all rhombuses, F4 the set of all squares and F5 the set of trapeziums in a plane. Then F1 may be equal to

(a) ${F}_{2}\cap {F}_{3}$                        (b) ${F}_{3}\cap {F}_{4}$                        (c) ${F}_{2}\cup {F}_{3}$                       (d) ${F}_{2}\cup {F}_{3}\cup {F}_{4}\cup {F}_{1}$

We know that every rectangle, rhombus and square in a plane is a parallelogram but every trapezium is not a parallelogram.

So, F1 is either of F1 or F2 or F3 or F4.

$\therefore {F}_{1}={F}_{1}\cup {F}_{2}\cup {F}_{3}\cup {F}_{4}$

Hence, the correct answer is option (d).

#### Question 1:

Describe the following sets in Roster form:
(i) {x : x is a letter before e in the English alphabet};
(ii) {xN : x2 < 25};
(iii) {xN : x is a prime number, 10 < x < 20};
(iv) {xN : x = 2n, nN};
(v) {xR : x > x}.
(vi) {x : x is a prime number which is a divisor of 60}
(vii) {x : x is a two digit number such that the sum of its digits is 8}
(viii) The set of all letters in the word 'Trigonometry'
(ix) The set of all letters in the word 'Better'.

Roster form:
In this form, a set is defined by listing elements, separated by commas, within braces {}.
(i) {a, b, c, d}
(ii) {1, 2, 3, 4}
(iii) {11, 13, 17, 19}
(iv) {2, 4, 6, 8, 10,...}
(v) $\mathrm{\varphi }$
(vi) {2, 3, 5}
(vii) {17, 26, 35, 44, 53, 62, 71, 80}
(viii) {T, R, I, G, O, N, M, E, Y}
(ix) {B, E, T, R}

#### Question 2:

Describe the following sets in set-builder form:
(i) A = {1, 2, 3, 4, 5, 6};
(ii)
(iii) C = {0, 3, 6, 9, 12, ...};
(iv) D = {10, 11, 12, 13, 14, 15};
(v) E = {0};
(vi) {1, 4, 9, 16, ..., 100}
(vii) {2, 4, 6, 8 .....}
(viii) {5, 25, 125 625}

Set-builder form:
To describe a set, a variable x (each element of the set) is written inside braces. Then, after putting a colon, the common property P(x) possessed by each element of the set is written within braces.

#### Question 3:

List all the elements of the following sets:
(i)

(ii)

(iii)

(iv) D = {x : x is a vowel in the word "EQUATION"}

(v) E = {x : x is a month of a year not having 31 days}

(vi) F = {x : x is a letter of the word "MISSISSIPPI"}

#### Question 4:

Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form:

 (i) {A, P, L, E} (i) x : x + 5 = 5, x ∈ Z (ii) {5, −5} (ii) {x : x is a prime natural number and a divisor of 10} (iii) {0} (iii) {x : x is a letter of the word "RAJASTHAN"} (iv) {1, 2, 5, 10,} (iv) {x: x is a natural number and divisor of 10} (v) {A, H, J, R, S, T, N} (v) x : x2 − 25 = 0 (vi) {2, 5} (vi) {x : x is a letter of the word "APPLE"}

(i) {A, P, L, E} is a roster form of {x : x is a letter of the word APPLE}.
(ii) {5, −5} is a roster form of {x : x2 − 25 = 0}.
(iii) {0} is a roster form of  {x : x + 5 = 5, x ∈ Z}.
(iv) {1, 2, 5, 10} is a roster form of {x : x is a natural number and a divisor of 10}.
(v) {A, H, J, R, S, T, N} is a roster form of {x : x is a letter of the word RAJASTHAN}.
(vi) {2, 5} is a roster form of {x : x is a prime natural number and a divisor of 10}.

 (i) {A, P, L, E} (vi) {x : x is a letter of the word APPLE} (ii) {5, −5} (v) { x : x2 − 25 = 0} (iii) {0} (i) {x : x + 5 = 5, x ∈ Z} (iv) {1, 2, 5, 10} (iv) {x : x is a natural number and a divisor of 10} (v) {A, H, J, R, S, T, N} (iii) {x : x is a letter of the word RAJASTHAN} (vi) {2, 5} (ii) {x : x is a prime natural number and a divisor of 10}

#### Question 5:

Write the set of all vowels in the English alphabet which precede q.

The set of vowels in the English alphabet that precede q is {a, e, i, o}.

#### Question 6:

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

The set of all positive integers whose cube is odd is {2n + 1 : n$\in$Z, n$\ge$0}.

#### Question 7:

Write the set in the set-builder form.

The set-builder form of the set is $\left\{\frac{n}{{n}^{2}+1}:n\in N,n\le 7\right\}$.

#### Question 1:

Which of the following are examples of empty set?
(i) Set of all even natural numbers divisible by 5;
(ii) Set of all even prime numbers;
(iii) {x : x2 −2 = 0 and x is rational};
(iv) {x : x is a natural number, x < 8 and simultaneously x > 12};
(v) {x : x is a point common to any two parallel lines}.

(i) All natural numbers that end with 0 are even & divisible by 5. Therefore, the given set is not an example of empty set.
(ii) 2 is an even prime number. Therefore, the given set is not an example of empty set.
(iii) There is no rational number whose square is 2 such that x2$-$2 = 0. Therefore, it is example of empty set.
(iv) It is not possible that x$<$8 and, at the same time, x$>12$. Therefore, it is an example of empty set.
(v) There is no common point in two parallel lines. Therefore, it is an example of empty set.

#### Question 2:

Which of the following sets are finite and which are infinite?
(i) Set of concentric circles in a plane;
(ii) Set of letters of the English Alphabets;
(iii) {xN : x > 5};
(iv) {x = ∈ N : x < 200};
(v) {xZ : x < 5};
(vi) {xR : 0 < x < 1}.

(i) There can be infinite concentric circles in a plane. Therefore, it is an infinite set.
(ii) There are 26 letters in the set of English alphabet. Therefore, it is a finite set.
(iii) {xN : x > 5} = {6,7,8,9,...}. There will be infinite numbers. So, it an infinite set.
(iv) There are finite elements in the set {x = ∈ N : x < 200}. Therefore, it is a finite set.
(v) In this set, xZ , so there would be infinite elements in the set {xZ : x < 5}. Therefore, it is an infinite set.
(vi) In this set,  x ∈ R. We know real numbers include all numbers, i.e., decimal numbers, rational numbers and irrational numbers.
So, there would be infinite elements in the set {xR : 0 < x < 1}. Therefore, it is an infinite set.

#### Question 3:

Which of the following sets are equal?
(i)
(ii)
(iii)
(iv) .

Two sets A & B are equal if every element of A is a member of B & every element of B is a member of A.
(i)
(ii)
Set B would be {1}.
(iii)
It can be written as {1, 2, 3} because we do not repeat the elements while writing the elements of a set.
C = {1, 2, 3}
(iv) includes elements {1, 2, 3}.
∴ D = {1, 2, 3}
Hence, we can say that A = C = D.

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