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**Which of the following are sets? ****Justify your answer.****Q.1. ****(a) ****The collection of all the months of a year beginning with letter M****(b) The collection of difficult topics in Mathematics. Ans.**

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(i) ∉

(ii) ∈

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A = {× : × ϵ Z, - 1/2 < × < 11/2}

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(a) Empty

(b) Non-empty

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A = {x : x is a letter in the word FOLLOW}

B = {y : y is a letter in the word WOLF}

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A= {F, O, L, W}

B = {W, O, L, F}

Hence A = B

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Hence {{3, 4}}

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**Q.****22. Write the set in roster form A = The set of all letters in the word T R I G N O M E T R Y Ans.** A = {T, R, I, G, N, O, M, E, Y}

A, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”.

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B = {L, O, Y, A}

Hence A = B

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Hence {3, 4} ϵ A is correct and

{3, 4} ⊂ A is incorrect

(i) A∪A' = --------

(ii) ( A')' = ---------

(iii) A∩A' = --------

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(i) U

(ii) A

(iii) ∅

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x

x

x(x - 5) +3(x - 5) = 0

(x - 5) (x + 3) = 0

x = 5

x = -3 [x = -3 reject]

∴ x = 5

E = {5}

Hence, C = E

Accordingly,

n(A)=21,n(B)=26,n(C)=29,n(A∩B)=14,n(C∩A)=12,

n(B∩C)=14,n(A∩B∩C)=8

The Venn diagram for the given problem can be drawn as above.

It can be seen that number of people who like product C only is

=29–(4+8+6)=11**Q.****30. A college awarded 38 medals in football, 15 in basketball and 20 in cricket. If these medals went to a total of 50 men and only five men got medals in all the three sports, how many received medals in exactly two of the three sports? ****Ans. **People who got medals in exactly two of the three sports.**Hint:**

a, b, c, d, e, f, g – Number of elements in bounded region

f = 5

a + b + f + e = 38

b + c + d + f = 15

e + d + f + g = 20

a + b + c + d + e + f + g = 50

we have to find b + d + e**Q.****31. There are 200 individuals with a skin disorder, 120 had been exposed to the chemical C _{1}, 50 to chemical C_{2}, and 30 to both the chemicals C_{1} and C_{2}. Find the number of individuals exposed to**

(1) chemical C_{1} but not chemical C_{2}

(2) chemical C_{2} but not chemical C_{1}

(3) chemical C_{1} or chemical C

n(U) = 200, n(A) = 120, n(B) = 50, n(A∩B) = 30

= 120 - 30 = 90

= 50 - 30 = 20

= 120 + 50 - 30

= 140

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b + c + e + f = 26

c + d + f + g = 29

b + c = 14,c + f = 15, c + d = 12

c = 8

d = 4, c = 8, f = 7, b = 6, g = 10, e = 5, a = 3

People who like product c only = g = 10

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n (A) = 38, n (B) = 15, n (C) = 20

n (A ∪ B ∪ C) = 58 and n (A ∩ B ∩ C) = 3

n (A ∪ B ∪ C) = n (A) + n (B) + n(C) – n (A ∩ B) – n (B ∩ C) – n (C ∩ A) + n (A ∩ B ∩ C)

58 = 38 + 15 + 20 – (a + d ) – (d + c) – (b + d) + 3

18 = a + d + c + b + d

18 = a + b + c + 3d

18 = a + b + c + 3 * 3

9 = a + b + c

b + c + e + f = 26

c + d + f +g = 26

c + d = 9

b + c = 11

c + f = 8

c = 3

f = 5, b = 8, d = 6, c = 3, g = 12

e = 10, a = 8

(i) a + b + c + d + e + f + g = 52

(ii) a + e + g = 30

(i) Two classes meet at the same hour

(ii) The two classes met at different hours and ten students are enrolled in both the courses.

n (C) = 20, n (P) = 30

(i) C ∩ P = ϕ ⇒ n(C∩P) = 0

n (C ∪ P) = n (C) + n (P) + n (C ∩ P)

= 20 + 30 + 0

= 50

(ii) n (C ∩ P) = 10

n (C ∪ F) = n (C) + n (F) - n (C ∩ P)

= 20 + 30 - 10

= 40

(i) only chemistry

(ii) only mathematics

(iii) only physics

(iv) physics and chemistry but mathematics

(v) mathematics and physics but not chemistry

(vi) only one of the subjects

(vii) at least one of three subjects

(viii) None of three subjects.

n (P) = b + c + e + f = 1 2

n(C) = d + e + f + g = 11

n (M ∩ P) = b + e =9

n (M ∩ C) = d + e = 5

n (P ∩ C) = e + f = 4

e = 3

so b = 6, d = 2, f = 1

a = 4, g = 5, c = 2

(i) g = 5

(ii) a = 4

(iii) c = 2

(iv) f = 1

(v) b = 6

(vi) g + a + c = 11

(vii) a + b + c + d + e + f + g + = 23

(viii) 25 - (a + b + c + d + e + f + g) = 25 - 23 = 2

(i) How many students were studying Hindi?

(ii) How many students were studying English and Hindi?

a + e = 23, e + g = 8

a + e + g + d = 26

e + g + f + c = 48

g + f = 8

so, e = 5, g = 3, d = 0, f = 5, c = 35

(i) d + g + f + b = 0 + 3 + 5 + 10 = 18

(ii) d + g = 0 + 3 = 3

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