Samacheer Kalvi Class 9 Maths Solutions Chapter 1 Set Language Exercise 1.3

Question 1. Using the given venn diagram, write the elements of
(i) A
(ii) B
(iii) A∪B
(iv) A∩B
(v) A – B
(vi) B - A
(vii) A'
(viii) B'
(ix) U
Answer:
(i) \( A = \{2, 4, 7, 8, 10\} \)
(ii) \( B = \{3, 4, 6, 7, 9, 11\} \)
(iii) \( A \cup B = \{2, 3, 4, 6, 7, 8, 9, 10, 11\} \)
(iv) \( A \cap B = \{4, 7\} \)
(v) \( A \setminus B = \{2, 8, 10\} \)
(vi) \( B \setminus A = \{3, 6, 9, 11\} \)
(vii) \( A' = \{1, 3, 6, 9, 11, 12\} \)
(viii) \( B' = \{1, 2, 8, 10, 12\} \)
(ix) \( U = \{1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12\} \)

In simple words: We identified each element in the universal set and then sorted them into their correct places within sets A, B, their union, intersection, and complements based on the provided Venn diagram. Every element is distinct and has a specific position in the diagram.

🎯 Exam Tip: Carefully observe each region of the Venn diagram to ensure all elements are correctly assigned to their respective sets or areas. Pay attention to the elements that are common to both sets and those that are outside both sets.

Question 2. Find AUB, AnB, A – B and B – A for the following sets
(i) A = {2, 6, 10, 14} and B = {2, 5, 14, 16}
(ii) A = {a, b, c, e, u} and B = {a, e, i, o, u}
(iii) A = {x : x ∈ N, x ≤ 10} and B = {x : x ∈ W, x < 6}
(iv) A = Set of all letters in the word “mathematics” and B = Set of all letters in the word "geometry"
Answer:
(i) For \( A = \{2, 6, 10, 14\} \) and \( B = \{2, 5, 14, 16\} \):
\( A \cup B = \{2, 5, 6, 10, 14, 16\} \)
\( A \cap B = \{2, 14\} \)
\( A \setminus B = \{6, 10\} \)
\( B \setminus A = \{5, 16\} \)

(ii) For \( A = \{a, b, c, e, u\} \) and \( B = \{a, e, i, o, u\} \):
\( A \cup B = \{a, b, c, e, i, o, u\} \)
\( A \cap B = \{a, e, u\} \)
\( A \setminus B = \{b, c\} \)
\( B \setminus A = \{i, o\} \)

(iii) First, list the elements for the sets:
\( A = \{x : x \in N, x \leq 10\} \implies A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
\( B = \{x : x \in W, x < 6\} \implies B = \{0, 1, 2, 3, 4, 5\} \)
Now, perform the operations:
\( A \cup B = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
\( A \cap B = \{1, 2, 3, 4, 5\} \)
\( A \setminus B = \{6, 7, 8, 9, 10\} \)
\( B \setminus A = \{0\} \)

(iv) First, list the distinct letters for the sets:
\( A = \{m, a, t, h, e, i, c, s\} \)
\( B = \{g, e, o, m, t, r, y\} \)
Now, perform the operations:
\( A \cup B = \{a, c, e, g, h, i, m, o, r, s, t, y\} \)
\( A \cap B = \{e, m, t\} \)
\( A \setminus B = \{a, c, h, i, s\} \)
\( B \setminus A = \{g, o, r, y\} \)
In simple words: We found the elements in each set by listing them out. Then, for each pair of sets, we combined all unique elements to find the union, found the common elements for the intersection, and identified elements present in one set but not the other for the differences. The order of elements in a set does not change the set itself.

🎯 Exam Tip: Remember that set operations like union, intersection, and difference deal with unique elements. When listing elements, ensure there are no duplicates. For sets defined by rules, carefully interpret the rule to list all elements correctly.

Question 3. If U = {a, b, c, d, e, f, g, h}, A = {b, d, f, h} and B = {a, d, e, h}, find the following sets.
(i) A'
(ii) B'
(iii) A'∪B'
(iv) A'∩B'
(v) (A∪B)'
(vi) (A∩B)'
(vii) (A')'
(viii) (B')'
Answer:
Given:
\( U = \{a, b, c, d, e, f, g, h\} \)
\( A = \{b, d, f, h\} \)
\( B = \{a, d, e, h\} \)

(i) To find \( A' \):
\( A' = U \setminus A \)
\( = \{a, b, c, d, e, f, g, h\} \setminus \{b, d, f, h\} \)
\( = \{a, c, e, g\} \)

(ii) To find \( B' \):
\( B' = U \setminus B \)
\( = \{a, b, c, d, e, f, g, h\} \setminus \{a, d, e, h\} \)
\( = \{b, c, f, g\} \)

(iii) To find \( A' \cup B' \):
\( A' \cup B' = \{a, c, e, g\} \cup \{b, c, f, g\} \)
\( = \{a, b, c, e, f, g\} \)

(iv) To find \( A' \cap B' \):
\( A' \cap B' = \{a, c, e, g\} \cap \{b, c, f, g\} \)
\( = \{c, g\} \)

(v) To find \( (A \cup B)' \):
First, find \( A \cup B \):
\( A \cup B = \{b, d, f, h\} \cup \{a, d, e, h\} \)
\( = \{a, b, d, e, f, h\} \)
Next, find \( (A \cup B)' \):
\( (A \cup B)' = U \setminus (A \cup B) \)
\( = \{a, b, c, d, e, f, g, h\} \setminus \{a, b, d, e, f, h\} \)
\( = \{c, g\} \)

(vi) To find \( (A \cap B)' \):
First, find \( A \cap B \):
\( A \cap B = \{b, d, f, h\} \cap \{a, d, e, h\} \)
\( = \{d, h\} \)
Next, find \( (A \cap B)' \):
\( (A \cap B)' = U \setminus (A \cap B) \)
\( = \{a, b, c, d, e, f, g, h\} \setminus \{d, h\} \)
\( = \{a, b, c, e, f, g\} \)

(vii) To find \( (A')' \):
From (i), \( A' = \{a, c, e, g\} \)
\( (A')' = U \setminus A' \)
\( = \{a, b, c, d, e, f, g, h\} \setminus \{a, c, e, g\} \)
\( = \{b, d, f, h\} \)
This is equal to set \( A \).

(viii) To find \( (B')' \):
From (ii), \( B' = \{b, c, f, g\} \)
\( (B')' = U \setminus B' \)
\( = \{a, b, c, d, e, f, g, h\} \setminus \{b, c, f, g\} \)
\( = \{a, d, e, h\} \)
This is equal to set \( B \).
In simple words: We performed various set operations like finding complements, unions, and intersections. A complement means all elements in the universal set that are not in the given set. We also saw that the complement of a complement brings us back to the original set. This demonstrates the basic rules of set theory.

🎯 Exam Tip: When dealing with complements, always remember the universal set. De Morgan's Laws, like \( (A \cup B)' = A' \cap B' \) and \( (A \cap B)' = A' \cup B' \), are important for verifying your answers for compound complements.

Question 4. Let U = {0, 1, 2, 3, 4, 5, 6, 7} A = {1, 3, 5, 7} and B = {0, 2, 3, 5, 7}, find the following sets.
(i) A'
(ii) B'
(iii) A'∪B'
(iv) A'∩B'
(v) (A∪B)'
(vi) (A∩B)'
(vii) (A')'
(viii) (B')'
Answer:
Given:
\( U = \{0, 1, 2, 3, 4, 5, 6, 7\} \)
\( A = \{1, 3, 5, 7\} \)
\( B = \{0, 2, 3, 5, 7\} \)

(i) To find \( A' \):
\( A' = U \setminus A \)
\( = \{0, 1, 2, 3, 4, 5, 6, 7\} \setminus \{1, 3, 5, 7\} \)
\( = \{0, 2, 4, 6\} \)

(ii) To find \( B' \):
\( B' = U \setminus B \)
\( = \{0, 1, 2, 3, 4, 5, 6, 7\} \setminus \{0, 2, 3, 5, 7\} \)
\( = \{1, 4, 6\} \)

(iii) To find \( A' \cup B' \):
\( A' \cup B' = \{0, 2, 4, 6\} \cup \{1, 4, 6\} \)
\( = \{0, 1, 2, 4, 6\} \)

(iv) To find \( A' \cap B' \):
\( A' \cap B' = \{0, 2, 4, 6\} \cap \{1, 4, 6\} \)
\( = \{4, 6\} \)

(v) To find \( (A \cup B)' \):
First, find \( A \cup B \):
\( A \cup B = \{1, 3, 5, 7\} \cup \{0, 2, 3, 5, 7\} \)
\( = \{0, 1, 2, 3, 5, 7\} \)
Next, find \( (A \cup B)' \):
\( (A \cup B)' = U \setminus (A \cup B) \)
\( = \{0, 1, 2, 3, 4, 5, 6, 7\} \setminus \{0, 1, 2, 3, 5, 7\} \)
\( = \{4, 6\} \)

(vi) To find \( (A \cap B)' \):
First, find \( A \cap B \):
\( A \cap B = \{1, 3, 5, 7\} \cap \{0, 2, 3, 5, 7\} \)
\( = \{3, 5, 7\} \)
Next, find \( (A \cap B)' \):
\( (A \cap B)' = U \setminus (A \cap B) \)
\( = \{0, 1, 2, 3, 4, 5, 6, 7\} \setminus \{3, 5, 7\} \)
\( = \{0, 1, 2, 4, 6\} \)

(vii) To find \( (A')' \):
From (i), \( A' = \{0, 2, 4, 6\} \)
\( (A')' = U \setminus A' \)
\( = \{0, 1, 2, 3, 4, 5, 6, 7\} \setminus \{0, 2, 4, 6\} \)
\( = \{1, 3, 5, 7\} \)
This is equal to set \( A \).

(viii) To find \( (B')' \):
From (ii), \( B' = \{1, 4, 6\} \)
\( (B')' = U \setminus B' \)
\( = \{0, 1, 2, 3, 4, 5, 6, 7\} \setminus \{1, 4, 6\} \)
\( = \{0, 2, 3, 5, 7\} \)
This is equal to set \( B \).
In simple words: We applied basic set operations like complements, unions, and intersections to numerical sets. It is important to work step-by-step, first finding the elements of intermediate sets (like \( A \cup B \) or \( A \cap B \)) before calculating their complements. The universal set helps define the boundary for all elements.

🎯 Exam Tip: Pay close attention to the definition of the universal set (U) as it dictates the elements available for complement operations. Always list the elements clearly before performing any set operations to avoid errors.

Question 5. Find the symmetric difference between the following sets.
(i) P = {2, 3, 5, 7, 11} and Q = {1, 3, 5, 11}
(ii) R = {l, m, n, o, p} and S = {j, I, n, q}
(iii) X = {5, 6, 7} and Y = {5, 7, 9, 10}
Answer:
The symmetric difference of two sets A and B, denoted as \( A \Delta B \), is the set of elements that are in either A or B, but not in their intersection. It can be found using the formula \( A \Delta B = (A \setminus B) \cup (B \setminus A) \) or \( A \Delta B = (A \cup B) \setminus (A \cap B) \).

(i) For \( P = \{2, 3, 5, 7, 11\} \) and \( Q = \{1, 3, 5, 11\} \):
Method 1: Using \( (P \cup Q) \setminus (P \cap Q) \)
\( P \cup Q = \{2, 3, 5, 7, 11\} \cup \{1, 3, 5, 11\} \)
\( = \{1, 2, 3, 5, 7, 11\} \)
\( P \cap Q = \{2, 3, 5, 7, 11\} \cap \{1, 3, 5, 11\} \)
\( = \{3, 5, 11\} \)
\( P \Delta Q = (P \cup Q) \setminus (P \cap Q) \)
\( = \{1, 2, 3, 5, 7, 11\} \setminus \{3, 5, 11\} \)
\( = \{1, 2, 7\} \)

Method 2: Using \( (P \setminus Q) \cup (Q \setminus P) \)
\( P \setminus Q = \{2, 3, 5, 7, 11\} \setminus \{1, 3, 5, 11\} \)
\( = \{2, 7\} \)
\( Q \setminus P = \{1, 3, 5, 11\} \setminus \{2, 3, 5, 7, 11\} \)
\( = \{1\} \)
\( P \Delta Q = (P \setminus Q) \cup (Q \setminus P) \)
\( = \{2, 7\} \cup \{1\} \)
\( = \{1, 2, 7\} \)

(ii) For \( R = \{l, m, n, o, p\} \) and \( S = \{j, l, n, q\} \):
Method 1: Using \( (R \cup S) \setminus (R \cap S) \)
\( R \cup S = \{l, m, n, o, p\} \cup \{j, l, n, q\} \)
\( = \{j, l, m, n, o, p, q\} \)
\( R \cap S = \{l, m, n, o, p\} \cap \{j, l, n, q\} \)
\( = \{l, n\} \)
\( R \Delta S = (R \cup S) \setminus (R \cap S) \)
\( = \{j, l, m, n, o, p, q\} \setminus \{l, n\} \)
\( = \{j, m, o, p, q\} \)

Method 2: Using \( (R \setminus S) \cup (S \setminus R) \)
\( R \setminus S = \{l, m, n, o, p\} \setminus \{j, l, n, q\} \)
\( = \{m, o, p\} \)
\( S \setminus R = \{j, l, n, q\} \setminus \{l, m, n, o, p\} \)
\( = \{j, q\} \)
\( R \Delta S = (R \setminus S) \cup (S \setminus R) \)
\( = \{m, o, p\} \cup \{j, q\} \)
\( = \{j, m, o, p, q\} \)

(iii) For \( X = \{5, 6, 7\} \) and \( Y = \{5, 7, 9, 10\} \):
Method 1: Using \( (X \cup Y) \setminus (X \cap Y) \)
\( X \cup Y = \{5, 6, 7\} \cup \{5, 7, 9, 10\} \)
\( = \{5, 6, 7, 9, 10\} \)
\( X \cap Y = \{5, 6, 7\} \cap \{5, 7, 9, 10\} \)
\( = \{5, 7\} \)
\( X \Delta Y = (X \cup Y) \setminus (X \cap Y) \)
\( = \{5, 6, 7, 9, 10\} \setminus \{5, 7\} \)
\( = \{6, 9, 10\} \)

Method 2: Using \( (X \setminus Y) \cup (Y \setminus X) \)
\( X \setminus Y = \{5, 6, 7\} \setminus \{5, 7, 9, 10\} \)
\( = \{6\} \)
\( Y \setminus X = \{5, 7, 9, 10\} \setminus \{5, 6, 7\} \)
\( = \{9, 10\} \)
\( X \Delta Y = (X \setminus Y) \cup (Y \setminus X) \)
\( = \{6\} \cup \{9, 10\} \)
\( = \{6, 9, 10\} \)
In simple words: The symmetric difference finds all the elements that are unique to each set, ignoring any elements they share. You can think of it as finding what's in 'A only' plus what's in 'B only'. It is a way to see all elements that are not common between two sets.

🎯 Exam Tip: Always state the formula you are using for symmetric difference before applying it. Both \( (A \setminus B) \cup (B \setminus A) \) and \( (A \cup B) \setminus (A \cap B) \) are valid, so choose the one you find easier to calculate and ensure consistency.

Question 6. Using the set symbols, write down the expressions for the shaded region in the following
Answer:
(i) The shaded region represents the elements that are in set Y but not in set X. This is written as \( Y \setminus X \).

(ii) The shaded region represents all elements that are neither in set X nor in set Y. This is the complement of their union, written as \( (X \cup Y)' \).

(iii) The shaded region represents elements that are in set X only, or in set Y only, but not in their common part. This is the symmetric difference, written as \( (X \setminus Y) \cup (Y \setminus X) \) or \( X \Delta Y \).

In simple words: Venn diagrams use shaded areas to show the results of set operations like difference, complement, or symmetric difference. Understanding what each shaded region means is key to writing the correct set expression. Visual representations make set operations easier to understand.

🎯 Exam Tip: When identifying shaded regions, mentally or physically trace the boundaries of the sets. Remember that \( A \setminus B \) is "A only," \( A' \) is "outside A," and \( A \Delta B \) is "A only OR B only."

Question 7. Let A and B be two overlapping sets and the universal set U. Draw appropriate Venn diagram for each of the following,
(i) A∪B
(ii) A∩B
(iii) (A∩B)'
(iv) (B – A)'
(v) A'∪B'
(vi) A'∩B'
(vii) What do you observe from the Venn diagram (iii) and (v)?
Answer:
(i) \( A \cup B \)

(ii) \( A \cap B \)

(iii) \( (A \cap B)' \)

(iv) \( (B \setminus A)' \)

(v) \( A' \cup B' \)

(vi) \( A' \cap B' \)

(vii) What do you observe from the Venn diagram (iii) and (v)?
From the diagrams, we observe that \( (A \cap B)' = A' \cup B' \). Both diagrams show the exact same shaded region, which includes everything except the intersection of A and B.
In simple words: We drew Venn diagrams for different set operations like union, intersection, and complements. Drawing these diagrams helps to clearly see which elements are included in the result of each operation. Visualizing set operations with Venn diagrams clearly demonstrates important rules, such as De Morgan's Laws.

🎯 Exam Tip: Practice drawing various Venn diagrams to become familiar with different shaded regions for common set operations. Remember De Morgan's Laws: \( (A \cup B)' = A' \cap B' \) and \( (A \cap B)' = A' \cup B' \), as these are often tested.