- Introduction
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- Methods of representing a set
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- Types of sets
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- Subset and Proper Subset
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- Subset of set of the real numbers
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- Interval as subset of R Real Number
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- Power Set
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- Universal Set
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- Venn diagram
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- Operation on Sets

**Fill in the blank**

(a) If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, Then A ∪ B is ……….

(b) The sets A and B are having elements 10 and 8,n(A ∩ B) =2, the n(A ∪B) is …..

(c) A set Z contains 4 elements, and then the number of elements in the Power set of Z will be……

(d) The set Z={x: x^{2} -3=0,x is a rational number) is an……….set

(1) Set of odd natural numbers divisible by 2 is a null set

(2) Set of even prime numbers is not an null set

(3) {*x*:*x *is a natural numbers, *x *< 4 and *x *> 11 } is an infinite set

(4) {*y*:*y *is a point common to any two parallel lines} is an infinite set

(5) The set of months of a year is a finite set

(6) {0,1, 2, 3 ...} is a finite set

(7) {1, 2, 3 ... 999} is an infinite set

(8) The set of positive integers greater than 99 is an infinite set

(9) The set of lines which are parallel to the *y*-axis is an infinite set

(10) The set of letters in the English alphabet is a finite set

(11) The set of natural numbers under 200 which are multiple of 7 is finite set

(12) The set of animals living on the earth is a finite set

(13) The set of circles passing through the origin (0, 0) is a finite set

(14) The set A = {-2, -3}; B = {*x*: *x *is solution of *x*2 + 5*x *+ 6 = 0} are equal sets

(15) The set P = {*x*: *x *is a letter in the word FOLLOW}; Q = {*y*: *y *is a letter in the word WOLF} are not equal sets

(16) {2, 3, 4} ⊂ {1, 2, 3, 4, 5}

(17) {*a*, *b*, *c*}⊄ {*b*, *c*, *d*}

(18) {*x*: *x *is a student of Class X of your school} ⊄ {*x*: *x *student of your school}

(19) {*x*: *x *is a square in the plane} ⊄ {*x*: *x *is a rectangle in the same plane}

(20) {*p*: *p *is a triangle in a plane} ⊂ {*p*: *p *is a rectangle in the plane}

(21) {*x*: *x *is an equilateral triangle in a plane} ⊂ {*x*: *x *is a triangle in the same plane}

(22) {*y*: *y *is an odd natural number} ⊂ {*y*: *y *is an integer}

(23) {*a*, *b*} ⊄ {*b*, *c*, *a*}

(24) {*a*, *e*} ⊂ {*p*: *p *is a vowel in the English alphabet}

(25) {1, 2, 3} ⊂{1, 3, 5}

(26) {*p*} ⊂ {*p,* *q*, *s*}

(27) {*a*} ∈ (*1*, *2*, *3*)

(28) {*x*: *x *is an even natural number less than 6} ⊂ {*x*: *x *is a natural number which divides 36}

(29) If *x *∈ A and A ∈ B, then *x *∈ B

(30) If A ⊂ B and B ∈ C, then A ∈ C

(31) If A ⊂ B and B ⊂ C, then A ⊂ C

(32) If A ⊄ B and B ⊄ C, then A ⊄ C

(33) If *x *∈ A and A ⊄ B, then *x *∈ B

(34) If A ⊂ B and *x *∉ B, then *x *∉ A

**Solutions**

- T
- T
- F
- F
- T
- F
- F
- T
- T
- T
- T
- T
- F
- T
- F
- T
- T
- F
- F
- F
- T
- T
- F
- T
- F
- T
- F
- T
- F
- F
- T
- F
- F
- T

**Subjective Questions**

Write the following sets in roster form:

**(1) **U = {*x*: *x *is an integer and –10 < *x *< 10}.

**(2) **V = {*x*: *x *is a natural number less than 18}.

**(3) **W = {*x*: *x *is a two-digit natural number such that the sum of its digits is 9}

**(4) **X = {*x*: *x *is a prime number which is divisor of 90}.

**(5) **Y= The set of all letters in the word MATHEMATICS.

**(6) **Z= The set of all letters in the word INTEGRATION.

**Linked Type comprehension**

If U = {1,2,3, 5, 7, 9, 11}, V = {7, 9, 11, 13}, W = {11, 13, 15} and X = {15, 17,19,21,23}; find

**(i) **U ∩ V

**(ii) **V ∩ W

**(iii) **U ∩ W ∩ X

**(iv) **U ∩ W

**(v) **V ∩ X

**(vi) **U ∩ (V ∪ W)

**(vii) **U ∩ X

**(viii) **U ∩ (V ∪ X)

**(ix) **(U ∩ V) ∩ (V ∪ W)

**(x) **(U ∪ X) ∩ (V ∪ W)

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20},

C = {2, 4, 6, 8, 10, 12, 14, 16}, 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

**(ix) **C – B

**(x) **D – B

**(xi) **C – D

**Subjective Questions**

- In a group of 400 people in USA, 250 can speak Spanish and 200 can speak English. How many people can speak both Spanish and English?

**Solution**

Let S be the set of people who speak Spanish, and

E be the set of people who speak English

∴ *n*(S ∪ E) = 400, *n*(S) = 250, *n*(E) = 200

*n*(S ∩ E) = ?

We know that:

*n*(S ∪ E) = *n*(S) + *n*(E) – *n*(S ∩ E)

∴ 400 = 250 + 200 – *n*(S ∩ E)

⇒ 400 = 450 – *n*(S ∩ E)

⇒ *n*(S ∩ E) = 450 – 400

∴ *n*(S ∩ E) = 50

Thus, 50 people can speak both Spanish and English.

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

**Solution**

It is given that:

*n(*P) = 40, *n*(P ∪ Q) = 60, *n*(P ∩ Q) = 10

We know that:

*n*(P ∪ Q) = *n*(P) + *n*(Q) – *n*(P ∩ Q)

∴ 60 = 40 + *n*(Q) – 10

∴ *n*(Q) = 60 – (40 – 10) = 30

Thus, the set Q has 30 elements.

**Subjective question on number of elements**

(a) U={x: x is positive integer less than 1000 and divisible by 7} , n(U) =?

(b) V={x: x is positive integer less than 1000 and divisible by 7 but not by 11} , n(V)=?

(c) P={x: x is positive integer less than 1000 and divisible by 7 and 11} , n(P)=?

(d) Q={x: x is positive integer less than 1000 and divisible by either 7 or 11} , n(Q)=?

(e) A={x: x is positive integer less than 1000 and divisible by exactly one of 7 or 11} , n(A)=?

(f ) B={x: x is positive integer less than 1000 and divisible by neither 7 nor 11} , n(B)=?

(g) C={x: x is positive integer less than 1000 and have distinct digits} , n(C)=?

(h) D={x: x is positive integer less than 1000 and have distinct digits and even} , n(D)=?

**Solutions**

- 142
- 130
- 12
- 220
- 208
- 779
- 738
- 373

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