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**1) Which of the following collections are sets? i) The collection of all prime numbers between 7 and 19. **

i) A natural number ‘p’ is called a prime number if it is greater than one and its only factors are ‘1’ and ‘p’.

Prime numbers between 7 and 19 are 11, 13, 17.

∴ Given collection is a set and contains 11, 13, 17.

ii) The concept of ‘rich person’ is vague and there is no rule for deciding whether a particular person is rich or not.

∴ Given collection is not a set.

**2) Which of the following collections are set? i) Collection of all factors of 50 which are greater than 6. ii) The collection of all integer which are not natural numbers.**

i) The factor of 50 are 1, 2, 5, 10, 25 and 50 and out of these, 10, 25 and 50 are greater than 6.

∴ Given collection is a set and contains 10, 25 and 50.

ii) Integers are 0, ±1, ±2, ±3…. And natural numbers are 1, 2, 3,…..

∴ Integers which are not natural numbers are 0, -1, -2, -3,…..

∴ Given collection is a set and contains 0, -1, -2, -3,…..

**3) Write the following sets by roster method:**** i) The set of all natural number ‘x’ such 4x + 9 < 50 **

ii) The set of all integer ‘x’ such that x^{2} + 5x + 6 = 0

i) 4x + 9 < 50 ⇒ 4x < 41⇒ x < 41/4 ⇒ x < 10.25

Since x is a natural number, so x can take values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

∴ Given set = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

ii)

-3, -2 are both integers .

Given set = {-3, -2}.

**4) Write the following set by roaster method: The set of all nature number ‘x’ such that |x| ≤ 4 **

Since, x is positive, we have |x| = x

∴ |x| ≤ 4 implies x ≤ 4

∴ The possible values of x are 1, 2, 3, 4.

∴ Given set = {1, 2, 3, 4}.**5) Describe the following set by set property method: **** {6, 10, 14, 18}**

The given set is {6, 10, 14, 18}

We observe that 6 = 4 (1) + 2,

10 = 4(2) + 2,

14 = 4(3) + 2

18 = 4(4) + 2

∴ The given set is equal to

{x : x = 4n + 2, n ≤ 4, n N}.

**6) Describe the following set by set property method: **** {2, 3, 5, 7, 11}**

The given set is {2, 3, 5, 7, 11}.

We observe that 2, 3, 5, 7, 11 are all prime numbers.

∴ Given set = {x:x is a prime number, x ≤ 11}.

**7) Describe the following set by set property method: **** {1/4, 1/8. 1/16, 1/32, 1/64}**

The given set is {1/4, 1/8. 1/16, 1/32, 1/64}

We observe that 1/4= 1/2^{2}, 1/8 = 1/2^{3}, 1/16 = 1/2^{4}, 1/32 = 1/2^{5}, 1/64 = 1/2^{6}

∴ The elements of the given set are of the for, 1/2^{n}, where n = 2, 3, 4, 5, 6.

∴ Given set = {x:x = 1/2^{n}, n N, 2 ≤ n ≤ 6}.

**8) Describe the following sets by roster method: **** i) {x : x ^{2} + 5x + 6 = 0, x N} **

i) The equation is x

∴ x = -2, -3

But -2, -3 are not in N. ∴ Given set = {}

ii) x

-2, -4 are in Z. ∴ Given set = {-2, -4}.

**9) Describe the following sets by roster method: **** i) {x : 4x + 7 < 25, x N} **** ii) the set of all letters in the word. **** TRIGONOMETRY.**

i) 4x + 7

ii) Different letters in the word TRIGONOMETRY are T, R, I, G, O, N, M, E, Y.

∴ Given set = {T, R, I, G, O, N, M, E, Y}.

**10) Which of the following sets are null sets?**** i) The set A of all prime numbers lying between 15 and 19. **** ii) A = {x : x is a natural number between 1 and 2 }**

i) Natural numbers lying between 15 and 19 are 16, 17, 18. Out of these, only 17 is a prime number.

∴ Given set = {17}, which is not a null set.

ii) There is no natural number between 1 and 2.

∴ Given set is a null set.

**11) Which of the following sets are null sets? **** i) A = {x : x ^{2} = 16, x N}**

i) x

∴ Given set {4}, which is not a null set.

ii) The numerical value |x| of any number x cannot be negative.

∴ |x| < - 4 is not true for any number

∴ Given set is a null set.

**12) Which of the following sets are singleton sets? **** i) The set of all prime numbers less than 3. **** ii) {x : |x| = 7, x N}**

i) Natural numbers less than 3 are 1 and 2.

By definition, 2 is the first prime number.

∴ Given set = {2}, which is a singleton

ii) |x| = 7 implies x = ± 7

x = -7 does not belong to N, where as x = 7 belong to the set.

∴ Given set = {7}, which is a singleton.

**13) Which of the following sets are singleton sets? **** i) {x : x ^{2} + 2x + 1 = 0, x N} **

ii) {x : x

i) x

x + 1 = 0 ⇒ x = -1

x = -1 is not a natural number

∴ Given set = {}, which is not a singleton set.

ii) x

|x| ≤ 3 implies x = -3, -2, -1, 0, 2, 3.

∴ x = -3, 3 satisfies x

∴ Given set = {3}, which is a singleton set.

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