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Looking for NCERT Class 11 Maths Exercise 1.4 solutions? You’re at the right destination! This section provides comprehensive and step-by-step solutions to all the questions in Exercise 1.4 of Chapter 1 – Sets. Designed as per the NCERT curriculum, these solutions cover important topics like complement of a set, practical problems on union and intersection, and more. Whether you’re preparing for an upcoming test or simply want to strengthen your understanding of sets, these Class 11 Chapter 1 Maths solutions are perfect for you. Download or go through the solutions now and gain confidence in tackling set theory problems!
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Exercise 1.4
1. Find the union of each of the following pairs of sets:
(i) \( x=\{1,3,5\} y=\{1,2,3\} \)
Answer
It is given in the question that,
\(X=\{1,3,5\}\)
And, \( y=\{1,2,3\} \)
All the members of both sets together are \( 1,3,5,1,2,3 \). Now we don't have to replicate members. So, the members of union of these two sets are \( 1,2,3,5 \)
\( \therefore X U Y\{1,2,3,5\} \)
(ii) \( \mathrm{A}=[\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\} \ \mathrm{B}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\} \)
Answer
It is given in the question that,
\( A=\{a, e, i, o, u\} \)
And, \( \mathrm{B}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\} \) All the members of both sets together are \( \mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u} \), \( a, b, c \) Now we don't have to replicate members. So, the members of union of these two sets are \( \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u} \)
\( \therefore A U B\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{e}, \mathrm{I}, \mathrm{o}, \mathrm{u}\} \)
(iii) \( \mathrm{A}=\{x: x \) is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}
B = {x : x is a natural number less than 6}
Answer
It is given in the question that,
\(A=\{3,6,9,12,15,18, \ldots \ldots\}\)
And, \( B=\{1,2,3,4,5 \)
\( \text{All the members of both sets together are }3,6,9,12,15,18, \ldots \ldots \ldots \ldots, 1 \), \( 2,3,4,5 \)
Now we don't have to replicate members. So, the members of union of these two sets are \( 1,2,3,4,5,6,9,12,15,18 \), \( \qquad \)
\( \therefore \) A U B \( \{1,2,3,4,5,6,9,12,15,18, \ldots\} \)
(iv) \( \mathrm{A}=\{x: x \) is a natural number and 1 < x ≤ 6}
B = {x : x is a natural number and 6 < x < 10}
B = {x : x is a natural number and 6 < x < 10}
Answer
It is given in the question that,
\(A=\{2,3,4,5,6\}\)
And, \( B=\{7,8,9\} \)
All the members of both sets together are \( 2,3,4,5,6,7,8,9 \).
Now we don't have to replicate members. So, the members of union of these two sets are \( 2,3,4,5,6,7,8,9 \)
\( \therefore \mathrm{A}UB\{2,3,4,5,6,7,8,9\} \)
(v) \( \mathrm{A}=\{1,2,3\}, \mathrm{B}=\phi \)
Answer
It is given in the question that,
\( \mathrm{A}=\{1,2,3\} \)
And, \( \mathrm{B}=\phi \)
All the members of both sets together are 1, 2, 3
Now we don't have to replicate members. So, the members of union of these two sets are \( 1,2,3 \)
\( \therefore AUB \{1,2,3\} \)
2. Let \( A=\{a, b\}, B=\{a, b, c\} \). Is \( A \subset B \) ? What is \( A \cup B \) ?
Answer
It is given in the question that,
\(A=\{a, b\}\)
And, \( \mathrm{B}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\} \)
Here, it is clearly seen that all the elements of set A are present in set B
3. If \( A \) and \( B \) are two sets such that \( A \subset B \), then what is \( A \cup B \) ?
Answer
It is given in the question that, \( A \) and \( B \) are two sets such that,
\( \mathrm{A} \subset \mathrm{B} \)
Let us take, \( \mathrm{A}=\{\mathrm{a}, \mathrm{b}\} \)
And, \( B=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\} \) we have:
\( A \cup B=\{a, b, c\}=B \)
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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) \( \mathrm{A} \cup \mathrm{B} \)
Answer
It is given in the question that,
\(\mathrm{A}=\{1,2,3,4\}\)
And, \( B=\{3,4,5,6\} \)
\( \therefore \mathrm{A} \cup \mathrm{B} \{1,2,2,4,5,6\} \)
(ii) \( \mathrm{A} \cup \mathrm{C} \)
Answer
It is given in the question that,
\(\mathrm{A}=\{1,2,3,4\}\)
And, \( \mathrm{C}=\{5,6,7,8\} \)
\(\therefore \mathrm{A} \cup \mathrm{C} \{1,2,3,4,5,6,7,8,\}\)
(iii) \( \mathrm{B} \cup \mathrm{C} \)
Answer
It is given in the question that,
\(\mathrm{B}=\{3,4,5,6\}\)
And, \( \mathrm{C}=\{5,6,7,8\} \)
\(\therefore B \cup C\{3,4,5,6,7,8\}\)
(iv) \( \mathrm{B} \cup \mathrm{D} \)
Answer
It is given in the question that,
\(B=\{3,4,5,6\}\)
And, \( \mathrm{D}=\{7,8,9,10\} \)
\(\therefore \mathrm{B} \cup \mathrm{D}\{3,4,5,6,7,8,9,10\}\)
(v) \( A \cup B \cup C \)
Answer
It is given in the question that,
\(\mathrm{A}=\{1,2,3,4\}\)
\(\mathrm{B}=\{3,4,5,6\}\)
And, \( \mathrm{C}=\{5,6,7,8\} \)
\(\therefore A \cup B \cup C\{1,2,3,4\} \cup\{3,4,5,6,\} \cup\{5,6,7,8\}=\{1,2,3,4,5,6,7\)
8}
(vi) \( A \cup B \cup D \)
Answer
It is given in the question that,
\(A=\{1,2,3,4\}\)
\(B=\{3,4,5,6\}\)
And, \( \mathrm{D}=\{7,8,9,10\} \)
\(\therefore A \cup B \cup D\{1,2,3,4\} \cup\{3,4,5,6\} \cup\{7,8,9,10\}=\{1,2,3,4,5,6,7 \text {, }\)
\(8,9,10\}\)
(vii) \( B \cup C \cup D \)
Answer
It is given in the question that,
\(B=\{3,4,5,6\}\)
\(C=\{5,6,7,8\}\)
And, \( \mathrm{D}=\{7,8,9,10\} \)
\(\therefore B \cup C \cup D\{3,4,5,6\} \cup\{5,6,7,8\} \cup\{7,8,9,10\}=\{3,4,5,6,7,8,9 \text {, }\)
\(10\}\)
5. Find the intersection of each pair of sets of question
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(i) \( x=\{1,3,5\} y=\{1,2,3\} \)
Answer
It is given in the question that,
\(X=\{1,3,5\}\)
And, \( y=\{1,2,3\} \)
\( \therefore x \cap y=\{1,3\} \)
(ii) \( \mathrm{A}=[\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\} \mathrm{B}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\} \)
Answer
It is given in the question that,
\( A=\{a, e, i, o, u\} \)
And, \( B=\{a, b, c\} \)
\( \therefore \mathrm{A} \cap \mathrm{B}=\{\mathrm{a}\} \)
(iii) A = {x: x is a natural number and multiple of 3}
B = {x: x is a natural number less than 6}
B = {x: x is a natural number less than 6}
Answer
It is given in the question that,
\(A=\{3,6,9,12,15,18, \ldots \ldots\}\)
And, \( \mathrm{B}=\{1,2,3,4,5\} \)
\( \therefore \mathrm{A} \cap \mathrm{B}=\{3\} \)
(iv) A = {x: x is a natural number and 1 < x ≤ 6}
B = {x: x is a natural number and 6 < x < 10}
B = {x: x is a natural number and 6 < x < 10}
Answer
It is given in the question that,
\(A=\{2,3,4,5,6\}\)
And, \( B=\{7,8,9\} \)
\( \therefore \mathrm{A} \cap \mathrm{B}=\{\phi\} \)
(v) \( \mathrm{A}=\{1,2,3\}, \mathrm{B}=\phi \)
Answer
It is given in the question that,
\(\mathrm{A}=\{1,2,3\}\)
And, \( \mathrm{B}=\phi \)
\( \therefore \mathrm{A} \cap \mathrm{B}=\{\phi\} \)
6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find:
(i) \( \mathrm{A} \cap \mathrm{B} \)
Answer
It is given in the question that,
\( A=\{3,5,7,9,11\} \)
And, \( B=\{7,9,1,13\} \)
\( \therefore \mathrm{A} \cap \mathrm{B}=\{3,5,7,9,11\} \cap\{7,9,11,13\} \)
\( \mathrm{A} \cap \mathrm{B} \) will give the members of set A and B that are common.
(ii) \( \mathrm{B} \cap \mathrm{C} \)
Answer
It is given in the question that,
\( B=\{7,9,11,13\} \)
And, \( \mathrm{C}=\{11,13,15\} \)
\( \therefore \mathrm{B} \cap \mathrm{C}=\{7,9,11,13\} \cap\{11,13,15\} \)
Intersection of two sets give the members of set that are common to both sets.
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(iii) \( A \cap C \cap D \)
Answer
It is given in the question that,
\( \mathrm{A}=\{3,5,7,9,11\} \)
\( \mathrm{C}=\{11,13,15\} \)
And, \( \mathrm{D}=\{15,17\} \)
\( \therefore \mathrm{A} \cap \mathrm{C} \cap \mathrm{D}=\{3,5,7,9,11,\} \cap\{11,13,15\} \cap\{15,17\} \)
As there are no members of sets that are common, the intersection of A , \( B \) and \( C \) will be a null set.
\( =\phi \)
(iv) \( \mathrm{A} \cap \mathrm{C} \)
Answer
It is given in the question that,
\( \mathrm{A}=\{3,5,7,9,11\} \)
And, \( \mathrm{C}=\{11,13,15\} \)
\( \therefore \mathrm{A} \cap \mathrm{C}=\{3,5,7,9,11\} \cap\{11,13,15\} \)
Only 11 is common to sets A and C. Therefore, \( =\{11\} \)
(v) \( B \cap D \)
Answer
It is given in the question that,
\(\mathrm{B}=\{7,9,11,13\}\)
And, \( \mathrm{D}=\{15,17\} \)
\( \therefore \mathrm{B} \cap \mathrm{D}=\{7,9,11,13\} \cap\{15,17\} \)
No member of set \( B \) is common with set \( D \), therefore intersection of and D will be a null set.
\(=\phi\)
(vi) \( A \cap(B \cup C) \)
Answer
It is given in the question that,
\(A=\{3,5,7,9,11\}\)
\(B=\{7,9,11,13\}\)
And, \( \mathrm{C}=\{11,13,15\} \)
\( \therefore \mathrm{A} \cap \mathrm{B} \cup \mathrm{C}=\{3,5,7,9,11\} \cap\{7,9,11,13\} \cup\{11,13,15\} \)
\( B \cup C \) will give all the members of both sets combined and not replicating the common members of the set.
After that intersection of set A and \(B \cup C\) is calculated as the members that are common to these two sets.
\(=\{3,5,7,9,11\} \cap\{7,9,11,13,15,17\}\)
\(=\{7,9,11\}\)
(vii) \( A \cap D \)
Answer
It is given in the question that,
\(\mathrm{A}=\{3,5,7,9,11\}\)
And, \( \mathrm{D}=\{15,17\} \)
\(\therefore \mathrm{A} \cap \mathrm{D}=\{3,5,7,9,11\} \cap\{15,17\}\)
No member of the two sets is common, therefore the intersection of these two sets will be a null set.
\(=\phi\)
(viii) \( A \cap(B \cup D) \)
Answer
It is given in the question that,
\(\mathrm{A}=\{3,5,7,9,11\}\)
\(\mathrm{B}=(7,9,11,13\}\)
And, \( \mathrm{D}=\{15,17\} \)
\(\therefore A \cap(B \cup D)=\{3,5,7,9,11\} \cap(\{7,9,11,13,15\} \cup\{15,17\})\)
\(\{3,5,7,9,11\} \cap\{7,9,11,13,15,17\}\)
\(= \{7,9,11\}\)
(ix) \( (A \cap B) \cap(B \cup C) \)
Answer
It is given in the question that,
\(A=\{3,5,7,9,11\}\)
\(B=(7,9,11,13\}\)
And, \( \mathrm{C}=\{11,13,15\} \)
\(\therefore A \cap(B \cup C)=\{3,5,7,9,11\} \cap(\{7,9,11,13,15\} \cup\{11,13,15\} 0\)
\(=\{7,9,11\}\)
(x) \( (\mathrm{A} \cup \mathrm{D}) \cap(B \cup C) \)
Answer
It is given in the question that,
\(\mathrm{A}=\{3,5,7,9,11\}\)
\(\mathrm{B}=(7,9,11,13\}\)
And, \( \mathrm{D}=\{15,17\} \)
Calculate A U B and B U C first and then intersection of these two sets is calculated \( =\{7,9,11,15\} \)
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7. If \( \mathrm{A}=\{x: x \) is a natural number \( \}, \mathrm{B}=\{x: x \) is an even natural number \(\}\)
\( \mathrm{C}=\{x: x \) is an odd natural number \( \} \) and \( \mathrm{D}=\{x: x \) is a prime number \( \} \), find:
(i) \( \mathrm{A} \cap \mathrm{B} \)
(ii) \( \mathrm{A} \cap \mathrm{C} \)
(iii) \( \mathrm{A} \cap \mathrm{D} \)
(iv) \( \mathrm{B} \cap \mathrm{C} \)
(v) \( B \cap D \)
(vi) \( \mathrm{C} \cap \mathrm{D} \)
\( \mathrm{C}=\{x: x \) is an odd natural number \( \} \) and \( \mathrm{D}=\{x: x \) is a prime number \( \} \), find:
(i) \( \mathrm{A} \cap \mathrm{B} \)
(ii) \( \mathrm{A} \cap \mathrm{C} \)
(iii) \( \mathrm{A} \cap \mathrm{D} \)
(iv) \( \mathrm{B} \cap \mathrm{C} \)
(v) \( B \cap D \)
(vi) \( \mathrm{C} \cap \mathrm{D} \)
Answer
\( \mathrm{A}=\{x: x \) is a natural number \( \}=\{1,2,3,4,5, \ldots\} \)
\( B=\{x: x \) is an even natural number \( \}=\{2,4,6,8, \ldots\} \)
\( \mathrm{C}=\{x: x \) is an odd natural number \( \}=\{1,3,5,7,9, \ldots\} \)
\( \mathrm{D}=\{x: x \) is a prime number \( \}=\{2,3,5,7, \ldots\} \)
(i) \( =A \cap B=\{x: x \) is an even natural number \( \}=B \)
(ii) \( \mathrm{A} \cap \mathrm{C}=\{x: x \) is an odd natural number \( \}=\mathrm{C} \)
(iii) \( \mathrm{A} \cap \mathrm{D}=\{x: x \) is a prime number \( \}=\mathrm{D} \)
(iv) \( \mathrm{B} \cap \mathrm{C}=\phi \)
(v) \( \mathrm{B} \cap \mathrm{D}=\{2\} \)
(vi) \( \mathrm{C} \cap \mathrm{D}=\{x: x \) is odd prime number \( \} \)
8. Which of the following pairs of sets are disjoint?
(i) \( \{1,2,3,4\} \) and \( \{x: x \) is a natural number and \( 4 \leq x \leq 6\} \)
(ii) \( \{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\} \) and \( \{\mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\} \)
(iii) \( \{x: x \) is an even integer \( \} \) and \( \{x: x \) is an odd integer \( \} \)
(i) \( \{1,2,3,4\} \) and \( \{x: x \) is a natural number and \( 4 \leq x \leq 6\} \)
(ii) \( \{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\} \) and \( \{\mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\} \)
(iii) \( \{x: x \) is an even integer \( \} \) and \( \{x: x \) is an odd integer \( \} \)
Answer
Disjoint sets are those sets which have no element in common. So, for that \( \mathrm{A} \cap \mathrm{B}=\phi \)
9. If \( A=\{3,6,9,12,15,18,21\}, B=\{4,8,12,16,20\} \),
\( \mathrm{C}=\{2,4,6,8,10,12,14,16\}, \mathrm{D}=\{5,10,15,20\} ; \) find:
(i) \( \mathrm{A}-\mathrm{B} \) (ii) \( \mathrm{A}-\mathrm{C} \) (iii) \( \mathrm{A}-\mathrm{D} \)
(iv) \( \mathrm{B}-\mathrm{A} \) (v) \( \mathrm{C}-\mathrm{A} \) (vi) \( \mathrm{D}-\mathrm{A} \)
(vii) \( \mathrm{B}-\mathrm{C} \) (viii) \( \mathrm{B}-\mathrm{D} \) (ix) \( \mathrm{C}-\mathrm{B} \)
(x) \( \mathrm{D}-\mathrm{B} \) (xi) C - D (xii) \( \mathrm{D}-\mathrm{C} \)
\( \mathrm{C}=\{2,4,6,8,10,12,14,16\}, \mathrm{D}=\{5,10,15,20\} ; \) find:
(i) \( \mathrm{A}-\mathrm{B} \) (ii) \( \mathrm{A}-\mathrm{C} \) (iii) \( \mathrm{A}-\mathrm{D} \)
(iv) \( \mathrm{B}-\mathrm{A} \) (v) \( \mathrm{C}-\mathrm{A} \) (vi) \( \mathrm{D}-\mathrm{A} \)
(vii) \( \mathrm{B}-\mathrm{C} \) (viii) \( \mathrm{B}-\mathrm{D} \) (ix) \( \mathrm{C}-\mathrm{B} \)
(x) \( \mathrm{D}-\mathrm{B} \) (xi) C - D (xii) \( \mathrm{D}-\mathrm{C} \)
Answer
A - B shows the relative complement of the set, which represents members of set A which are not part of set B
(i) \( \mathrm{A}-\mathrm{B}=\{3,6,9,15,18,21\} \)
(ii) \( \mathrm{A}-\mathrm{C}=\{3,9,15,18,21\} \)
(iii) \( \mathrm{A}-\mathrm{D}=\{3,6,9,12,18,21\} \)
(iv) \( \mathrm{B}-\mathrm{A}=\{4,8,16,20\} \)
(v) \( \mathrm{C}-\mathrm{A}=\{2,4,8,10,14,16\} \)
(vi) \( \mathrm{D}-\mathrm{A}=\{5,10,20\} \)
(vii) \( \mathrm{B}-\mathrm{C}=\{20\} \)
(viii) \( \mathrm{B}-\mathrm{D}=\{4,8,12,16\} \)
(ix) \( \mathrm{C}-\mathrm{B}=\{2,6,10,14\} \)
(x) \( \mathrm{D}-\mathrm{B}=\{5,10,15\} \)
(xi) \( \mathrm{C}-\mathrm{D}=\{2,4,6,8,12,14,16\} \)
(xii) \( \mathrm{D}-\mathrm{C}=\{5,15,20\} \)
10. If \( X=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\} \) and \( y=\{\mathrm{f}, \mathrm{b}, \mathrm{d}, \mathrm{g}\} \), find:
(i) \( x-y \)
(ii) \( y-x \)
(iii) \( x \cap y \)
(i) \( x-y \)
(ii) \( y-x \)
(iii) \( x \cap y \)
Answer
(i) \( x-y=\{\mathrm{a}, \mathrm{c}\} \)
(ii) \( y-x=\{\mathrm{f}, \mathrm{g}\} \)
(iii) \( x \cap y=\{\mathrm{b}, \mathrm{d}\} \)
11. If \( R \) is the set of real numbers and \( Q \) is the set of rational numbers, then what is \( \mathrm{R}-\mathrm{Q} \) ?
Answer
R: set of real numbers
Q : set of rational numbers
Therefore, \( \mathrm{R}-\mathrm{Q} \) is a set of irrational numbers.
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 \( 3 \in\{2,3,4,5\}, 3 \in\{3,6\} \)
\( \Rightarrow\{2,3,4,5\} \cap\{3,6\}=\{3\} \)
(ii) \( \{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\} \) and \( \{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\} \) are disjoint sets.
Answer
False
As \( \mathrm{a} \in\{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\}, \mathrm{a} \in\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\} \)
\( \Rightarrow\{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\} \cap\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}=\{\mathrm{a}\} \)
(iii) \( \{2,6,10,14\} \) and \( \{3,7,11,15\} \) are disjoint sets.
Answer
True
As \( \{2,6,10,14\} \cap\{3,7,11,15\}=\phi \)
(iv) \( \{2,6,10\} \) and \( \{3,7,11\} \) are disjoint sets.
Answer
True
As \( \{2,6,10\} \cap\{3,7,11\}=\phi \)
