Exercise 1.2 

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Q1. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) { x : x is a natural numbers, x < 5 and x > 7 }
(iv) { y : y is a point common to any two parallel lines}

Solution:

(i) There is any set of add number which is divisible by 2.

(ii) Let A is set which having even prime number

Therefore, A = {2}

So, this is not a null set

(iii) Let A = { x : x is a natural numbers, x < 5 and x > 7 }

therefore there is no natural numbers which is both x < 5 and x > 7

So, this is a null set. 

(iv) Two parallel lines never meet on a point, therefore this is a null set. 

Q2. Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3, . . .}
(iii) {1, 2, 3, . . .99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99

Solutions: 

(i) The set of months of a year is a finite set because it has 12 elements.

(ii) {1, 2, 3 ...} is an infinite set as it has infinite number of natural numbers.

(iii) {1, 2, 3 ...99, 100} has definite elements so it is a finite set as the numbers from 1 to 100 are finite in number.

(iv) The set of positive integers greater than 100 is an infinite set because positive integers greater than 100 are infinite in number.

(v)The set of prime numbers less than 99 is a finite set because prime numbers less than 99 are finite in number.

Q3. State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5

(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)

Solutions:

(i) Infinite: The set of lines which are parallel to the x-axis can be drawn infinitive in number.

Therefore, it is a infinite set.

(ii) Finite: The set of letters in the English alphabet is a finite set because English alphabet has only 26 number which is finite. 

(iii) Infinite: The set of numbers which are multiple of 5 is an infinite set because multiples of 5 are infinite in number.

(iv) Finite: Animals living on the earth is countable, therefore the set is finite.

(v) Infinite: Through the origin(0, 0) can be drawn infinite numbers of circle. Because many circle can be drawn through a point. 
Q4. In the following, state whether A = B or not:
(i) A = { a, b, c, d }

   B = { d, c, b, a }
(ii) A = { 4, 8, 12, 16 }

   B = { 8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}

    B = { x : x is positive even integer and x ≤ 10}
(iv) A = { x : x is a multiple of 10},

    B = { 10, 15, 20, 25, 30, . . . }

Solutions: 

(i) A = {a, b, c, d}; B = {d, c, b, a}

All elements of set A also belong to set B.

∴ A = B

(ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}

It can be seen that 12 ∈ A but 12 ∉ B.

∴ A ≠ B

(iii) A = {2, 4, 6, 8, 10}

B = {x: x is a positive even integer and x ≤ 10}

  = {2, 4, 6, 8, 10}

All elements of set A also belong to set B.

∴ A = B

(iv) A = {x: x is a multiple of 10}

B = {10, 15, 20, 25, 30 ...}

It can be seen that 15 ∈ B but 15 ∉ A.

∴ A ≠ B

Q5. Are the following pair of sets equal ? Give reasons.
(i) A = {2, 3}, B = {x : x is solution of x2 + 5x + 6 = 0}
(ii) A = { x : x is a letter in the word FOLLOW}
   B = { y : y is a letter in the word WOLF}

Solutions:

(i) A = {2, 3}; B = {x: x is a solution of x² + 5x + 6 = 0}

The equation x² + 5x + 6 = 0

=> x(x + 3) + 2(x + 3) = 0

=> (x + 2)(x + 3) = 0

=> x = –2 or x = –3

∴ A = {2, 3}; B = {–2, –3}

∴ A ≠ B

(ii) 

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

  = {F, O, L, W}

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

  = {W, O, L, F}

All elements of set A also belong to set B.

∴ A = B

Q6. From the sets given below, select equal sets :
A = { 2, 4, 8, 12},

B = { 1, 2, 3, 4},

C = { 4, 8, 12, 14},

D = { 3, 1, 4, 2}
E = {–1, 1},

F = { 0, a},

G = {1, –1},

H = { 0, 1}

Solution:

B and D are equal sets and also E and G are equal sets.