Problems & Solutions : Sets : Exercise 1(b) : Intermediate
Exercise 1(b) : ( problems & solutions ) :
1. Which of the following are examples of the null set.
(i) Set of odd natural numbers divisible by 2.
Solution :
The odd natural numbers are 1,3,5,...
Let A be the set of odd natural numbers divisible by 2.
We know no any odd number is divisible by 2.
∴ A has no elements.
Hence A is null set.
(ii) Set of even prime numbers.
Solution :
The prime numbers are 2,3,5,7,11,13,...
Let A be the set of even prime numbers.
Since 2 is only the even prime number, A = { 2 }
i.e. A has exactly one element 2.
Hence A is not null set.
(iii) { x : x is a natural number , x < 5 and x > 7 }
Solution :
Given set is { x : x is a natural number , x < 5 and x > 7 }.
= { 1,4,8,9,10,...}
Since the set containing elements, the given set is not null set.
(iv) { y : y is a point common to any two parallel lines }
Solution :
Given { y : y is a point common to any two parallel lines }
Since no any two parallel lines are intersect, there is no any point common to
any two parallel lines.
Hence the given set is null set.
2. Which of the following sets are finite or infinite ?
(i) The set of months of a year.
Solution :
Let A be the set of months of a year.
∴ A = { January, February, March, April, May, June, July, August, September, October,
November, December }.
Since the set A having exactly 12 elements, A is a finite set.
(ii) { 1,2,3,...}
Solution :
Given set is { 1 , 2 , 3 , ...}
The set has no end point.
∴ The given set is an infinite set.
(iii) { 1,2,3,...,99,100}
Solution :
Given set is { 1 , 2 , 3 , ..., 99 , 100 }
Since the given set has a last number ' 100 ' , the set is a finite set.
(iv) The set of positive integers greater than 100.
Solution :
Let A be the set of positive integers greater than 100.
∴ A = { 101,102,103,...}
Since A has no end point, A is an infinite set.
(v) The set of prime numbers less than 99.
Solution :
Let A be the set of prime numbers less than 99.
∴ A = { 2,3,5,7,11,...,97 }
Since A has a least and highest numbers , A is a finite set.
3. State whether each of the following set is finite or infinite :
(i) The set of lines which are parallel to the x-axis
Solution :
Let A be the set of lines which are parallel to the x-axis.
There are infinitely many lines which are parallel to x-axis.
∴ A is an infinite set.
(ii) The set of letters in English alphabet.
Solution :
Given the set of letters in English alphabet.
∴ The set is { A,B,C,D,...,X,Y,Z }
∴ The set has 26 alphabets i.e. 26 members.
Hence the set is finite.
(iii) The set of numbers which are multiples of 5.
Solution :
Let A be the set of numbers which are multiples of 5.
∴ A = { 5,10,15,20,25,30,...}
Since the multiples of 5 are infinite, A is infinite.
(iv) The set of animals living on the earth.
Solution :
Given the set of animals living on the earth.
Since the number of animals living on earth is finite, given set is finite.
(v) The set of circles passing through the origin (0,0).
Solution :
Given the set of circles passing through the origin.
Since there are infintely many circles passing through origin , the given
set is infinite.
3. In the following, state whether A = B or not:
(i) A = { a,b,c,d } B = { d,c,b,a }
Solution :
Given A = { a,b,c,d } & B = { d,c,b,a }
Since the elements of A and B are a,b,c,d , we have A = B.
(ii) A = { 4,8,12,16 } B = { 8,4,16,18 }
Solution :
Given A = { 4,8,12,16 } & B = { 8,4,16,18 }
The elements of A are 4,8,12,16
The elements of B are 4,8,16,18.
Since the elements of A and B are different, A ≠ B.
(iii) A = { 2,4,6,8,10} B = { x : x is positive even integer and x ≤ 10}
Solution :
Given A = { 2,4,6,8,10}
B = { x : x is positive even integer and x ≤ 10}
= { 2,4,6,8,10}
Since the elements of A and B are 2,4,6,8,10 , we have A = B.
(iv) A = { x : x is a multiple of 10 } , B = { 10,15,20,25,30,...}
Solution :
Given A = { x : x is a multiple of 10 }
= { 10,20,30,40,50,...}
Also given B = { 10,15,20,25,30,...}
Since the number 15 in B is not in A , A ≠ B.
5. Are the following pair of sets equal ? Give reasons.
(i) A = { 2,3 } and B = { x : x is a solution of x2+ 5x +6 = 0 }
Solution :
Given A = { 2,3 }
B = { x : x is a solution of x2+ 5x +6 = 0 }
We have x2+ 5x +6 = 0
⇒ x2+ 3x + 2x + 6 = 0
⇒ x ( x+3 ) + 2 ( x+3 ) = 0
⇒ (x+2) (x+3 ) = 0 .
⇒ x = -2,-3
∴ B = { x : x is a solution of x2+ 5x +6 = 0 }
= { -2,-3}
Since the numbers in A and B are different , A ≠ B.
(ii) A = { x : x is a letter in the word FOLLOW }
B = { x : x is a letter in the word WOLF }
Solution :
Given A = { x : x is a letter in the word FOLLOW }
= { F , O , L , W }
Also given
B = { x : x is a letter in the word WOLF }
= { W , O , L , F }
Since the words in both A and B are W , O , L , F , we have A = B.
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 :
Given
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 }
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