Exercise 1.4 NCERT Class 11 contains 12 questions. Each question contains several small questions. And each question of exercise 1.4 NCERT Class 11 is easy to solve if concepts are clear.
All previous exercises, like Exercise 1.1, Exercise 1.2, and Exercise 1.3, have already been completed.
Exercise 1.4 NCERT Class 11 Math Solutions
The first question of Exercise 1.4 NCERT Class 11 is about finding the union of sets of a given pair of sets. The union of sets A and B is the set which consists of all the elements of A and all the elements of B, the common elements being taken only once.
1. Find the union of each of the following pairs of sets:
(i) X = {1, 3, 5}
Y = {1, 2, 3}
Answer
The given sets are
X = {1, 3, 5}
Y = {1, 2, 3}
X ∪ Y is a set containing all elements of X and Y both.
So, X ∪ Y = {1, 2, 3, 5}. We write the common elements once.
(ii) A =
B =
Answer
The given sets are
A =
B =
The union of A and B is the set containing all elements of A and B.
So, A ∪ B =
(iii) A =
B =
Answer
The given sets in roster form are
A = {3, 6, 9, 12, 15, 18, 21, . . . }
B = {1, 2, 3, 4, 5}
The union of sets A and B is the collection of all elements of sets A and B.
So, A ∪ B = {1, 2, 3, 4, 5, 9, 12, 15, 18, 21, . . . }
We can write it in set-builder form as follows:
A ∪ B =
(iv) A =
B =
Answer
The roster form of the given sets are
A = {2, 3, 4, 5, 6}
B = {7, 8, 9}
The union of sets A and B is the collection of all numbers from 2 to 9.
So, A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}
It's set-builder form is given below:
A ∪ B =
Or
A ∪ B =
(v) A = {1, 2, 3}
B =
Answer
The given sets are
A = {1, 2, 3}
B =
Since B is an empty set, the union of A and B is the collection of elements of B only.
So, A ∪ B = {1, 2, 3}
The second, third, and fourth questions of Exercise 1.4 NCERT Class 11 are also about finding the union of sets.
2. Let A = , B = . Is A ⊂ B? What is A ∪ B?
Answer
The given sets are
A =
B =
Yes, A ⊂ B as all elements of A are in the set B.
A ∪ B =
3. If A and B are two sets such that A ⊂ B, then what is A ∪ B ?
Answer
A ⊂ B, it means all elements of A are in B also.
∴ A ∪ B = B
4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
(i) A ∪ B
Answer
The given sets A and B are
A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
∴ A ∪ B = {1, 2, 3, 4, 5, 6}
(ii) A ∪ C
Answer
A = {1, 2, 3, 4} and C = {5, 6, 7, 8}
∴ A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B ∪ C
Answer
B = {3, 4, 5, 6} and C = {5, 6, 7, 8}
∴ B ∪ C = {3, 4, 5, 6, 7, 8}
(iv) B ∪ D
Answer
B = {3, 4, 5, 6} and D = {7, 8, 9, 10}
∴ B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
(v) A ∪ B ∪ C
Answer
A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}
∴ A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(vi) A ∪ B ∪ D
Answer
A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and D = {7, 8, 9, 10}
∴ A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B ∪ C ∪ D
Answer
B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}
∴ B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
The fifth, sixth, and seventh questions of Exercise 1.4 are about finding the union and intersection of sets. The intersection of two sets A and B is the set of all elements that are common to both A and B. In other words, the intersection of two sets A and B is the set of all those elements that belong to both A and B.
5. Find the intersection of each of sets of question 1 above.
Answer
6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find
(i) A ∩ B
Answer
A = {3, 5, 7, 9, 11} and B = {7, 9, 11, 13}
∴ A ∩ B = {7, 9, 11}
(ii) B ∩ C
Answer
B = {7, 9, 11, 13} and C = {11, 13, 15}
∴ B ∩ C = {11, 13}
(iii) A ∩ C ∩ D
Answer
A = {3, 5, 7, 9, 11}, C = {11, 13, 15} and D = {15, 17}
∴ A ∩ C ∩ D = {} =
(iv) A ∩ C
Answer
A = {3, 5, 7, 9, 11} and C = {11, 13, 15}
∴ A ∩ C = {11}
(v) B ∩ D
Answer
B = {7, 9, 11, 13} and D = {15, 17}
∴ B ∩ D = {} =
(vi) A ∩ (B ∪ C)
Answer
A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}
First, we find
B ∪ C = {7, 9, 11, 13, 15}
and then
∴ A ∩ (B ∪ C) = {7, 9, 11}
(vii) A ∩ D
Answer
A = {3, 5, 7, 9, 11} and D = {15, 17}
∴ A ∩ D = {} =
(viii) A ∩ (B ∪ D)
Answer
Here, we have
A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and D = {15, 17}
B ∪ D = {7, 9, 11, 13, 15, 17}
∴ A ∩ (B ∪ D) = {7, 9, 11}
(ix) (A ∩ B) ∩ (B ∪ C)
Answer
A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}
Firstly, we calculate A ∩ B and B ∪ C, then (A ∩ B) ∩ (B ∪ C)
A ∩ B = {7, 9, 11}
B ∪ C = {7, 9, 11, 13, 15}
(A ∩ B) ∩ (B ∪ C) = {7, 9, 11}
(x) (A ∪ D) ∩ (B ∪ C)
Answer
A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}
Firstly, we calculate A U D and B U C,
A U D = {3, 5, 7, 9, 11, 15, 17}
B U C = {7, 9, 11, 13, 15}
(A U D) ∩ (B U C) = {7, 9, 11, 15}
7. If A = ,
B = ,
C = ,
D = , find
(i) A ∩ B
Answer
The given sets can be written in Roster form as follows:
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}
B = {2, 4, 6, 8, 10, . . .}
C = {1, 3, 5, 7, 9, 11, . . .}
D = {2, 3, 5, 7, 11, 13, . . .}
Since B is a set of even numbers and all even numbers are natural numbers
So, A ∩ B = B
(ii) A ∩ C
Answer
The sets in Roster form are
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}
and
C = {1, 3, 5, 7, 9, 11, . . .}
∵ C is a set of prime numbers, all prime numbers are natural numbers.
So, A ∩ C = C
(iii) A ∩ D
Answer
The given sets can be written in Roster form as follows:
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}
D = {2, 3, 5, 7, 11, 13, . . .}
Since all prime numbers are also natural numbers,
So, A ∩ D = D
(iv) B ∩ C
Answer
The sets B and C in roster form are
B = {2, 4, 6, 8, 10, . . .}
C = {1, 3, 5, 7, 9, 11, . . .}
Since, no odd natural number is an even number,
So, B ∩ C = {} = ϕ
(v) B ∩ D
Answer
The sets B and D in roster form are
B = {2, 4, 6, 8, 10, . . .}
D = {2, 3, 5, 7, 11, 13, . . .},
Since only 2 is the only even prime number,
So, B ∩ D = {2}
(vi) C ∩ D
Answer
The sets C and D in roster form are
C = {1, 3, 5, 7, 9, 11, . . .}
D = {2, 3, 5, 7, 11, 13, . . .}
Here, C ∩ D = {3, 5, 7, 11, 13, . . .}
C ∩ D is nothing but the set of odd prime numbers.
So, C ∩ D = {x : x is an odd prime number}
The eighth question of Exercise 1.4 NCERT Class 11 is about differentiating whether a set is disjoint or not. If two sets have no common elements, then the sets are called disjoint sets.
8. Which of the following pairs of sets are disjoint?
(i) {1, 2, 3, 4} and
(ii) and
(iii) and
Answer
(i) The sets are not disjoint as 4 belongs to both sets.
(ii) The sets are not disjoint as 'e' belongs to both sets.
(iii) The sets are disjoint as no even number is an odd number.
The ninth, tenth and eleventh questions of Exercise 1.4 NCERT Class 11 is about finding the difference of two sets. The difference of the sets A and B in this order is the set of elements which belongs to A but not to B.
9. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, and D = {5, 10, 15, 20}; find
(i) A − B
Answer
You can get the set A − B by removing all elements of B which are in A. In other words, by removing all common elements from the set A, we get the set A − B.
Here, A = {3, 6, 9, 12, 15, 18, 21}
B = {4, 8, 12, 16, 20}
So, A − B = {3, 6, 9, 15, 18, 21}
(ii) A − C
Answer
∵ A = {3, 6, 9, 12, 15, 18, 21}
C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ A − C = {3, 9, 15, 18, 21}
(iii) A − D
Answer
∵ A = {3, 6, 9, 12, 15, 18, 21}
D = {5, 10, 15, 20}
∴ A − D = {3, 6, 9, 12, 18, 21}
(iv) B − A
Answer
∵ A = {3, 6, 9, 12, 15, 18, 21}
B = {4, 8, 12, 16, 20}
∴ B − A = {4, 8, 16, 20}
(v) C − A
Answer
∵ A = {3, 6, 9, 12, 15, 18, 21}
C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ C − A = {2, 4, 8, 10, 14, 16}
(vi) D − A
Answer
∵ A = {3, 6, 9, 12, 15, 18, 21}
D = {5, 10, 15, 20}
∴ D − A = {5, 10, 20}
(vii) B − C
Answer
∵ B = {4, 8, 12, 16, 20}
C = {2, 4, 6, 8, 10, 12, 14, 16}
∴B − C = {} = ∅
(viii) B − D
Answer
∵ B = {4, 8, 12, 16, 20}
D = {5, 10, 15, 20}
∴ B − D = {4, 8, 12, 16}
(ix) C − B
Answer
∵ B = {4, 8, 12, 16, 20},
C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ C − B = {2, 6, 10, 14}
(x) D − B
Answer
∵ B = {4, 8, 12, 16, 20}
D = {5, 10, 15, 20}
∴ D − B = {5, 10, 15}
(xi) C − D
Answer
∵ C = {2, 4, 6, 8, 10, 12, 14, 16}
D = {5, 10, 15, 20}
∴ C − D = {2, 4, 6, 8, 12, 14, 16}
(xii) D − C
Answer
∵ C = {2, 4, 6, 8, 10, 12, 14, 16}
D = {5, 10, 15, 20}
∴ D − C = {5, 15, 20}
10. If X = and Y = , find
(i) X − Y
Answer
∵ X = and Y =
∴ X − Y =
(ii) Y − X
Answer
∵ X = and Y =
∴ Y − X =
(iii) X ∩ Y
Answer
∵ X = and Y =
∴ X ∩ Y =
11. If R is the set of real numbers and Q is the set of rational numbers, then what is R − Q?
Answer
We know that the set of real numbers R is actually union of rational numbers and irrational numbers.
Here, R = Set of real numbers
Q = Set of rational numbers
∴ R − Q = Set of irrational numbers
The twelfth question of Exercise 1.4 NCERT Class 11 is based on true/false questions on disjoint sets.
12. State whether each of the following statement is true or false. Justify your answer.
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
Answer
False, as the given sets {2, 3, 4, 5} and {3, 6} have a common element 3.
(ii) and are disjoint sets.
Answer
False, as the given sets and have a common element .
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
Answer
True, as the given sets {2, 6, 10, 14} and {3, 7, 11, 15} have no common elements.
(iii) {2, 6, 10} and {3, 7, 11} are disjoint sets.
Answer
True, as the given sets {2, 6, 10} and {3, 7, 11} have no common elements.
We have finished the solution of Exercise 1.4 NCERT Class 11. If you like the solutions, please let us know in the comments below about what you like, and do write us your invaluable recommendations of books that you need the solutions of.