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Exercise - 1.2

Question-1 :- 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)	A set of odd natural numbers divisible by 2 is a null set 
    because odd natural numbers are not divisible by 2. 

(ii) A set of even prime numbers is not a null set 
    because 2 is an even prime number. 

(iii) {x: x is a natural number, x < 5 and x > 7} is a null set 
    because a number cannot be simultaneously less than 5 and greater than 7. 

(iv) {y: y is a point common to any two parallel lines} is a null set 
    because two parallel lines do not have a common point. 
    

Question-2 :- 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

Solution :-
(i) The set of months of a year = finite set because {January, February,........December} 
(ii) {1, 2, 3 ...} = infinite set because last number is not given.
(iii) {1, 2, 3 ... 99, 100} = finite set because last number is given i.e., 100.
(iv) The set of positive integers greater than 100 = infinite set because we don't know about which 
     is last greater positive integer greater than 100
(v) The set of prime numbers less than 99 = finite set because {2, 3, 5,.....97}

Question-3 :- 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)

Solution :-
(i) The set of lines which are parallel to the x-axis is an infinite set 
   because lines parallel to the x-axis are infinite in number. 

(ii) The set of letters in the English alphabet is a finite set 
   because there are 26 letters in the English alphabet. 

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

(iv) The set of animals living on the earth is a finite set 
   because the number of animals living on the earth is every large but finite. 

(v)	The set of circles passing through the origin (0, 0) is an infinite set 
   because infinite number of circles can pass through the origin.  
    

Question-4 :- 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 ...}

Solution :-
(i) A = {a, b, c, d}; B = {d, c, b, a}
  The order in which the elements of a set are listed is not significant.
  Hence, A = B.

(ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18} 
  It can be seen that 12 ∈ A but 12 ∉ B.
  Hence, 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}
  Hence, 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.
  Hence, A ≠ B.

Question-5 :- 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}

Solution :-
(i)	A = {2, 3}; B = {x: x is solution of x² + 5x + 6 = 0} 
   x² + 5x + 6 = 0
   x² + 3x + 2x + 6 = 0
   x(x + 3) + 2(x + 3) = 0
   (x + 3)(x + 2) = 0
   x = -3, -2
   B = {-3, -2} 
   Hence, A ≠ B.
 
(ii) A = {x: x is a letter in the word FOLLOW}; B = {y: y is a letter in the word WOLF} 
   A = {F, O, L, W}; B = {W, O, L, F}
   Hence, A = B.
    

Question-6 :- 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 :-
  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} 
    
  It can be seen that
  Set B and D have same elements and also sets E and G have same element. 
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