GSEB Class 11 Maths Solutions Chapter 1 Sets Exercise 1.4

Question 1. Find the union of each of the following pairs of sets:
1. X = {1, 3, 5} Y = {1, 2, 3}
2. A = {a, e, i, o, u} B = {a, b, c}
3. A = {x : x is a natural number and multiple of 3}, B = {x : x is a natural number less than 6}.
4. A = {x : x is a natural number and 1 < x \( \leq \) 6}, B = {x: x is a natural number and 6 < x < 10}.
5. A = {1, 2, 3}, B = \( \phi \)
Answer:
1. \( X \cup Y = \{1, 3, 5\} \cup \{1, 2, 3\} = \{1, 2, 3, 5\} \)
2. \( A \cup B = \{a, e, i, o, u\} \cup \{a, b, c\} = \{a, b, c, e, i, o, u\} \)
3. \( A \cup B = \{3, 6, 9, ...\} \cup \{1, 2, 3, 4, 5\} = \{1, 2, 3, 4, 5, 6, 9, 12, 15, ...\} \)
4. \( A \cup B = \{2, 3, 4, 5, 6\} \cup \{7, 8, 9\} = \{2, 3, 4, 5, 6, 7, 8, 9\} \).
5. \( A \cup B = \{1, 2, 3\} \cup \phi = \{1, 2, 3\} \).
In simple words: To find the union of sets, you collect all unique elements from both sets. If an element appears in both, you only list it once.

Exam Tip: Remember that the union of two sets includes all elements that are in either set, without repeating any common elements.

Question 2. Let A = {a, b} and B = {a, b, c}. Is A \( \subset \) B? What is A \( \cup \) B?
Answer: Yes, \( A \subset B \), because every element of A is also an element of B. Therefore, A is a subset of B, i.e., \( A \subset B \).
\( A \cup B = \{a, b\} \cup \{a, b, c\} = \{a, b, c\} \).
In simple words: Yes, A is a subset of B because all things in A are also in B. When you join A and B, you get all items in B because B already has everything from A.

Exam Tip: A set P is a subset of set Q ( \( P \subset Q \) ) if every element of P is also an element of Q. The union of a set and its superset is always the superset.

Question 3. If A and B are two sets such that A \( \subset \) B, then what is A \( \cup \) B?
Answer: Since A is a subset of B, every element of set A is contained in set B, and hence \( A \cup B = B \).
In simple words: If set A is inside set B, then when you combine A and B, you will just get set B.

Exam Tip: Understand that if one set is a subset of another, their union will simply be the larger (superset) of the two sets.

Question 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 \( \cup \) B
(ii) A \( \cup \) C
(iii) B \( \cup \) C
(iv) B \( \cup \) D
(v) A \( \cup \) B \( \cup \) C
(vi) A \( \cup \) B \( \cup \) D
(vii) B \( \cup \) C \( \cup \) D
Answer:
(i) \( A \cup B = \{1, 2, 3, 4\} \cup \{3, 4, 5, 6\} = \{1, 2, 3, 4, 5, 6\} \)
(ii) \( A \cup C = \{1, 2, 3, 4\} \cup \{5, 6, 7, 8\} = \{1, 2, 3, 4, 5, 6, 7, 8\} \)
(iii) \( B \cup C = \{3, 4, 5, 6\} \cup \{5, 6, 7, 8\} = \{3, 4, 5, 6, 7, 8\} \)
(iv) \( B \cup D = \{3, 4, 5, 6\} \cup \{7, 8, 9, 10\} = \{3, 4, 5, 6, 7, 8, 9, 10\} \)
(v) \( A \cup B \cup C = (\{1, 2, 3, 4\} \cup \{3, 4, 5, 6\}) \cup \{5, 6, 7, 8\} \)
\( = \{1, 2, 3, 4, 5, 6\} \cup \{5, 6, 7, 8\} \)
\( = \{1, 2, 3, 4, 5, 6, 7, 8\} \).
(vi) \( A \cup B \cup D = (\{1, 2, 3, 4\} \cup \{3, 4, 5, 6\}) \cup \{7, 8, 9, 10\} \)
\( = \{1, 2, 3, 4, 5, 6\} \cup \{7, 8, 9, 10\} \)
\( = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).
(vii) \( B \cup C \cup D = (\{3, 4, 5, 6\} \cup \{5, 6, 7, 8\}) \cup \{7, 8, 9, 10\} \)
\( = \{3, 4, 5, 6, 7, 8\} \cup \{7, 8, 9, 10\} \)
\( = \{3, 4, 5, 6, 7, 8, 9, 10\} \).
In simple words: For each part, gather all the numbers from the sets mentioned. Make sure not to write any number more than once. When doing three sets, combine the first two, then combine that result with the third.

Exam Tip: When finding the union of multiple sets, it's often helpful to combine two sets at a time. List all unique elements from all involved sets.

Question 5. Find the intersection of each pair of sets of question 1 above?
Answer:
1. \( X \cap Y = \{1, 3, 5\} \cap \{1, 2, 3\} = \{1, 3\} \).
2. \( A \cap B = \{a, e, i, o, u\} \cap \{a, b, c\} = \{a\} \).
3. \( A \cap B = \{3, 6, 9, ...\} \cap \{1, 2, 3, 4, 5\} = \{3\} \).
4. \( A \cap B = \{2, 3, 4, 5, 6\} \cap \{7, 8, 9\} = \phi \).
5. \( A \cap B = \{1, 2, 3\} \cap \phi = \phi \).
In simple words: To find the intersection of sets, look for elements that appear in BOTH sets. If there are no common elements, the intersection is an empty set.

Exam Tip: Intersection means "what's common." If two sets have no elements in common, their intersection is the empty set, denoted by \( \phi \).

Question 6. If A = {3, 5, 7, 9, 11} and B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}, find:
(i) A \( \cap \) B
(ii) B \( \cap \) C
(iii) A \( \cap \) C \( \cap \) D
(iv) A \( \cap \) C
(v) B \( \cap \) D
(vi) A \( \cap \) (B \( \cup \) C)
(vii) A \( \cap \) D
(viii) A \( \cap \) (B \( \cup \) D)
(ix) (A \( \cap \) B) \( \cap \) (B \( \cup \) C)
(x) (A \( \cup \) D) \( \cap \) (B \( \cup \) C)
Answer:
(i) \( A \cap B = \{3, 5, 7, 9, 11\} \cap \{7, 9, 11, 13\} = \{7, 9, 11\} \)
(ii) \( B \cap C = \{7, 9, 11, 13\} \cap \{11, 13, 15\} = \{11, 13\} \).
(iii) \( A \cap C \cap D = (\{3, 5, 7, 9, 11\} \cap \{11, 13, 15\}) \cap \{15, 17\} \)
\( = \{11\} \cap \{15, 17\} = \phi \).
(iv) \( A \cap C = \{3, 5, 7, 9, 11\} \cap \{11, 13, 15\} = \{11\} \)
(v) \( B \cap D = \{7, 9, 11, 13\} \cap \{15, 17\} = \phi \).
(vi) First, find \( B \cup C \):
\( B \cup C = \{7, 9, 11, 13\} \cup \{11, 13, 15\} = \{7, 9, 11, 13, 15\} \)
Then, find \( A \cap (B \cup C) \):
\( A \cap (B \cup C) = \{3, 5, 7, 9, 11\} \cap \{7, 9, 11, 13, 15\} = \{7, 9, 11\} \).
(vii) \( A \cap D = \{3, 5, 7, 9, 11\} \cap \{15, 17\} = \phi \).
(viii) First, find \( B \cup D \):
\( B \cup D = \{7, 9, 11, 13\} \cup \{15, 17\} = \{7, 9, 11, 13, 15, 17\} \)
Then, find \( A \cap (B \cup D) \):
\( A \cap (B \cup D) = \{3, 5, 7, 9, 11\} \cap \{7, 9, 11, 13, 15, 17\} = \{7, 9, 11\} \).
(ix) First, find \( A \cap B \):
\( A \cap B = \{3, 5, 7, 9, 11\} \cap \{7, 9, 11, 13\} = \{7, 9, 11\} \)
Then, find \( B \cup C \):
\( B \cup C = \{7, 9, 11, 13\} \cup \{11, 13, 15\} = \{7, 9, 11, 13, 15\} \)
Then, find \( (A \cap B) \cap (B \cup C) \):
\( (A \cap B) \cap (B \cup C) = \{7, 9, 11\} \cap \{7, 9, 11, 13, 15\} = \{7, 9, 11\} \).
(x) First, find \( A \cup D \):
\( A \cup D = \{3, 5, 7, 9, 11\} \cup \{15, 17\} = \{3, 5, 7, 9, 11, 15, 17\} \)
Then, find \( B \cup C \):
\( B \cup C = \{7, 9, 11, 13, 15\} \)
Then, find \( (A \cup D) \cap (B \cup C) \):
\( (A \cup D) \cap (B \cup C) = \{3, 5, 7, 9, 11, 15, 17\} \cap \{7, 9, 11, 13, 15\} = \{7, 9, 11, 15\} \).
In simple words: Carefully apply the operations. For intersection, find common elements. For union, gather all elements. When there are brackets, solve inside them first, then do the outside operation.

Exam Tip: For expressions involving both union and intersection, remember to perform operations inside parentheses first. Pay close attention to the order of operations.

Question 7. Let A = {x: x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number}, D = {x : x is a prime number}. Find:
(i) A \( \cap \) B
(ii) A \( \cap \) C
(iii) A \( \cap \) D
(iv) B \( \cap \) C
(v) B \( \cap \) D
(vi) C \( \cap \) D
Answer:
Given:
\( A = \{1, 2, 3, 4, ...\} \)
\( B = \{2, 4, 6, 8, ...\} \)
\( C = \{1, 3, 5, 7, ...\} \)
\( D = \{2, 3, 5, 7, 11, 13, ...\} \)
(i) \( A \cap B = \{1, 2, 3, 4, ...\} \cap \{2, 4, 6, 8, ...\} \)
\( = \{2, 4, 6, 8, ...\} = B \).
(ii) \( A \cap C = \{1, 2, 3, 4, ...\} \cap \{1, 3, 5, 7, ...\} \)
\( = \{1, 3, 5, 7, ...\} = C \).
(iii) \( A \cap D = \{1, 2, 3, 4, ...\} \cap \{2, 3, 5, 7, 11, 13, ...\} \)
\( = \{2, 3, 5, 7, 11, 13, ...\} = D \).
(iv) \( B \cap C = \{2, 4, 6, 8, ...\} \cap \{1, 3, 5, 7, ...\} \)
\( = \phi \).
(v) \( B \cap D = \{2, 4, 6, 8, ...\} \cap \{2, 3, 5, 7, 11, 13, ...\} \)
\( = \{2\} \).
(vi) \( C \cap D = \{1, 3, 5, 7, ...\} \cap \{2, 3, 5, 7, 11, 13, ...\} \)
\( = \{3, 5, 7, 11, 13, ...\} \)
\( = \{x : x \text{ is an odd prime number}\} \).
In simple words: First, clearly write down what each set contains using dots for infinite sets. Then, find the common elements for each pair. If no common elements exist, the answer is an empty set.

Exam Tip: When dealing with infinite sets defined by properties (like natural, even, odd, prime numbers), it's useful to list a few elements to visualize them before finding intersections. Remember that 2 is the only even prime number.

Question 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) {a, e, i, o, u} and {c, d, e, f}
(iii) {x: x is an even integer} and {x : x is an odd integer}.
Answer:
(i) False, \( \{1, 2, 3, 4\} \cap \{4, 5, 6\} = \{4\} \neq \phi \). So, the given sets are not disjoint.
(ii) False, because \( \{a, e, i, o, u\} \cap \{a, b, c, d\} = \{a\} \neq \phi \). So, the given sets are not disjoint.
(iii) True, because \( \{x : x \text{ is an even integer}\} \cap \{x : x \text{ is an odd integer}\} = \phi \). Even and odd integers share no common elements, therefore they are disjoint sets.
In simple words: Two sets are disjoint if they do not share any elements. For each pair, check if there is anything in common. If yes, they are not disjoint. If no, they are.

Exam Tip: To determine if sets are disjoint, find their intersection. If the intersection is the empty set ( \( \phi \) ), then the sets are disjoint. Otherwise, they are not.

Question 9. Let 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
(ii) A - C
(iii) A - D
(iv) B - A
(v) C - A
(vi) D - A
(vii) B - C
(viii) B - D
(ix) C - B
(x) D - B
(xi) C - D
(xii) D - C
Answer:
(i) \( A - B = \{3, 6, 9, 12, 15, 18, 21\} - \{4, 8, 12, 16, 20\} \)
\( = \{3, 6, 9, 15, 18, 21\} \).
(ii) \( A - C = \{3, 6, 9, 12, 15, 18, 21\} - \{2, 4, 6, 8, 10, 12, 14, 16\} \)
\( = \{3, 9, 15, 18, 21\} \).
(iii) \( A - D = \{3, 6, 9, 12, 15, 18, 21\} - \{5, 10, 15, 20\} \)
\( = \{3, 6, 9, 12, 18, 21\} \).
(iv) \( B - A = \{4, 8, 12, 16, 20\} - \{3, 6, 9, 12, 15, 18, 21\} \)
\( = \{4, 8, 16, 20\} \).
(v) \( C - A = \{2, 4, 6, 8, 10, 12, 14, 16\} - \{3, 6, 9, 12, 15, 18, 21\} \)
\( = \{2, 4, 8, 10, 14, 16\} \).
(vi) \( D - A = \{5, 10, 15, 20\} - \{3, 6, 9, 12, 15, 18, 21\} \)
\( = \{5, 10, 20\} \).
(vii) \( B - C = \{4, 8, 12, 16, 20\} - \{2, 4, 6, 8, 10, 12, 14, 16\} \)
\( = \{20\} \).
(viii) \( B - D = \{4, 8, 12, 16, 20\} - \{5, 10, 15, 20\} \)
\( = \{4, 8, 12, 16\} \).
(ix) \( C - B = \{2, 4, 6, 8, 10, 12, 14, 16\} - \{4, 8, 12, 16, 20\} \)
\( = \{2, 6, 10, 14\} \).
(x) \( D - B = \{5, 10, 15, 20\} - \{4, 8, 12, 16, 20\} \)
\( = \{5, 10, 15\} \).
(xi) \( C - D = \{2, 4, 6, 8, 10, 12, 14, 16\} - \{5, 10, 15, 20\} \)
\( = \{2, 4, 6, 8, 12, 14, 16\} \).
(xii) \( D - C = \{5, 10, 15, 20\} - \{2, 4, 6, 8, 10, 12, 14, 16\} \)
\( = \{5, 15, 20\} \).
In simple words: Set difference (A - B) means taking all elements that are in set A but NOT in set B. You remove any common elements from the first set.

Exam Tip: When performing set difference (A - B), focus only on the elements present in A. Remove any of these elements that also appear in B. The elements unique to B are ignored.

Question 10. If X = {a, b, c, d} and Y = {f, b, d, g}, find:
1. X - Y
2. Y - X
3. X \( \cap \) Y
Answer:
1. \( X - Y = \{a, b, c, d\} - \{f, b, d, g\} = \{a, c\} \).
2. \( Y - X = \{f, b, d, g\} - \{a, b, c, d\} = \{f, g\} \).
3. \( X \cap Y = \{a, b, c, d\} \cap \{f, b, d, g\} = \{b, d\} \).
In simple words: For X-Y, list what's in X but not Y. For Y-X, list what's in Y but not X. For X \( \cap \) Y, list what both X and Y have in common.

Exam Tip: Set difference and intersection are distinct operations. Ensure you understand that A - B is generally not the same as B - A.

Question 11. If R is the set of real numbers and Q is the set of rational numbers, then what is R - Q?
Answer: Since the set of real numbers contains the set of rational numbers and the set of irrational numbers, therefore \( R - Q \) is the set of irrational numbers.
In simple words: Real numbers include both rational numbers and irrational numbers. If you take away all the rational numbers from the real numbers, you are left with just the irrational numbers.

Exam Tip: Remember the classification of number systems: Real numbers include all rational and irrational numbers. Rational numbers can be expressed as a fraction, while irrational numbers cannot.

Question 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.
(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.
Answer:
(i) False, \( \{2, 3, 4, 5\} \cap \{3, 6\} = \{3\} \neq \phi \). So, the given sets are not disjoint.
(ii) False, because \( \{a, e, i, o, u\} \cap \{a, b, c, d\} = \{a\} \neq \phi \). So, the given sets are not disjoint.
(iii) True, because \( \{2, 6, 10, 14\} \cap \{3, 7, 11, 15\} = \phi \). These sets have no elements in common.
(iv) True, because \( \{2, 6, 10\} \cap \{3, 7, 11\} = \phi \). These sets also share no elements.
In simple words: Check if any element is present in both sets. If you find even one common element, the sets are not disjoint. If no common elements are found, they are disjoint.

Exam Tip: Always look for common elements to decide if sets are disjoint. If the intersection is non-empty, the sets are not disjoint.