Samacheer Kalvi Class 9 Maths Solutions Chapter 1 Set Language Exercise 1.7

Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 1 Set Language Ex 1.7

I. Multiple Choice Questions.

Question 1. Which of the following is correct?
(a) \( \{7\} \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
(b) \( 1 \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
(c) \( 7 \notin \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
(d) \( \{7\} \subseteq \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
Answer: (b) \( 1 \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
In simple words: The symbol '\( \in \)' means "is an element of". Option (b) is correct because the number 1 is definitely one of the numbers listed inside the curly brackets. The set includes all whole numbers from 1 to 10.

🎯 Exam Tip: Remember the difference between 'element of' \( (\in) \) and 'subset of' \( (\subseteq) \). An element is a single item, while a subset is a set contained within another set.

Question 2. The set P = {x | x ∈ Z, -1 < x < 1} is a ...........
(a) Singleton set
(b) Power set
(c) Null set
(d) Subset
Answer: (a) Singleton set
In simple words: The set P contains all whole numbers (integers) between -1 and 1. The only integer that fits this description is 0. A set with exactly one element, like \( \{0\} \), is called a singleton set.

🎯 Exam Tip: Carefully read the conditions for 'x'. Integers (Z) include positive and negative whole numbers and zero, but not fractions or decimals. The strict inequalities \( < \) and \( > \) mean the endpoints are not included.

Question 3. If U = {x | x ∈ N, x < 10} and A = {x | x ∈ N, 2 ≤ x < 6} then (A')' is...........
(a) {1, 6, 7, 8, 9}
(b) {1, 2, 3, 4}
(c) {2, 3, 4, 5}
(d) { }
Answer: (c) {2, 3, 4, 5}
In simple words: First, \( U \) is the set of natural numbers less than 10, so \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Set \( A \) includes natural numbers from 2 up to, but not including, 6, making \( A = \{2, 3, 4, 5\} \). The complement of a complement, \( (A')' \), always brings you back to the original set \( A \).

🎯 Exam Tip: Remember that \( (A')' = A \). This property simplifies calculations a lot. Also, carefully list the elements for \( U \) and \( A \) based on the given conditions.

Question 4. If \( B \subset A \) then \( n(A \cap B) \) is...........
(a) \( n(A - B) \)
(b) \( n(B) \)
(c) \( n(B - A) \)
(d) \( n(A) \)
Answer: (b) \( n(B) \)
In simple words: When set B is a subset of set A, it means all elements of B are also in A. Therefore, the common elements between A and B (their intersection) will simply be all the elements in B. So, the number of elements in their intersection is the same as the number of elements in B.

🎯 Exam Tip: Visualize this concept: if B is inside A, then the overlapping part is just B itself. This simplifies the intersection to the smaller set B.

Question 5. If \( A = \{x, y, z\} \) then the number of non- empty subsets of A is..............
(a) 8
(b) 5
(c) 6
(d) 7
Answer: (d) 7
In simple words: The total number of subsets for a set with \( n \) elements is \( 2^n \). Since set A has 3 elements, it has \( 2^3 = 8 \) subsets. To find the number of non-empty subsets, we subtract 1 (for the empty set) from the total. So, \( 8 - 1 = 7 \).

🎯 Exam Tip: Always remember to subtract 1 from \( 2^n \) if the question asks for "non-empty" subsets, as the empty set \( \{\} \) is a subset of every set.

Question 6. Which of the following is correct?
(a) \( \emptyset \subseteq \{a, b\} \)
(b) \( \emptyset \in \{a, b\} \)
(c) \( \{a\} \in \{a, b\} \)
(d) \( a \subseteq \{a, b\} \)
Answer: (a) \( \emptyset \subseteq \{a, b\} \)
In simple words: The empty set \( (\emptyset) \) is a subset of every set, which makes option (a) correct. Option (b) is wrong because the empty set is not an *element* inside \( \{a, b\} \). Option (c) is wrong because \( \{a\} \) is a subset, not an element, of \( \{a, b\} \). Option (d) is wrong because an element \( a \) cannot be a subset; only a set can be a subset.

🎯 Exam Tip: Distinguish carefully between 'is an element of' \( (\in) \) and 'is a subset of' \( (\subseteq) \). The empty set is a subset of all sets, but it is an element only if it's explicitly listed inside the set's curly braces.

Question 7. If \( A \cup B = A \cap B \) then ...................
(a) \( A \neq B \)
(b) \( A = B \)
(c) \( A \subset B \)
(d) \( B \subset A \)
Answer: (b) \( A = B \)
In simple words: The union of two sets \( (A \cup B) \) includes all elements from both sets. The intersection \( (A \cap B) \) includes only the common elements. For these two to be equal, it can only happen if both sets contain exactly the same elements, meaning A and B are identical.

🎯 Exam Tip: Think of a Venn diagram. If the combined area (union) is the same as the overlapping area (intersection), the circles must be perfectly on top of each other, meaning the sets are equal.

Question 8. If \( B - A \) is \( B \), then \( A \cap B \) is.............
(a) A
(b) B
(c) U
(d) \( \emptyset \)
Answer: (d) \( \emptyset \)
In simple words: \( B - A \) means elements that are in B but not in A. If this operation results in the entire set B itself, it means there are no elements in B that are also in A. This can only happen if A and B have no common elements at all, which means their intersection is the empty set.

🎯 Exam Tip: When \( B - A = B \), it implies that A and B are disjoint sets, meaning they do not share any common elements. Disjoint sets always have an empty intersection.

Question 9. From the adjacent diagram \( n[P(A \Delta B)] \) is.............

🎯 Exam Tip: The symmetric difference \( A \Delta B \) can be found as \( (A \cup B) - (A \cap B) \) or \( (A - B) \cup (B - A) \). Once you have \( n(A \Delta B) \), calculate the power set size as \( 2^{n(A \Delta B)} \).

Question 10. If \( n(A) = 10 \) and \( n(B) = 15 \) then the minimum and maximum number of elements in \( A \cap B \) is ...........
(a) (10, 15)
(b) (15, 10)
(c) (10, 0)
(d) (0, 10)
Answer: (d) (0, 10)
In simple words: The maximum number of elements in \( A \cap B \) happens when one set is completely inside the other. In this case, if A is a subset of B, then \( A \cap B = A \), so \( n(A \cap B) = 10 \). The minimum number of elements in \( A \cap B \) happens when the sets have the largest possible union. The largest union is \( n(A) + n(B) = 10 + 15 = 25 \). Using the formula \( n(A \cap B) = n(A) + n(B) - n(A \cup B) \), the minimum intersection is \( 10 + 15 - 25 = 0 \). Thus, the minimum is 0 and the maximum is 10.

🎯 Exam Tip: To find the maximum intersection, consider the smaller set as entirely contained within the larger set. For the minimum intersection, consider the case where the union is maximized (i.e., \( n(A \cup B) = n(A) + n(B) \), if the universe allows it), which means the sets are as separate as possible.

Question 11. Let \( A = \{\emptyset\} \) and \( B = P(A) \) then \( A \cap B \) is...........
(a) \( \{\emptyset, \{0\}\} \)
(b) \( \{\emptyset\} \)
(c) \( \emptyset \)
(d) \( \{0\} \)
Answer: (b) \( \{\emptyset\} \)
In simple words: Set A contains one element, which is the empty set \( \emptyset \). Set B is the power set of A, meaning it contains all subsets of A. Since A has 1 element, its power set \( P(A) \) will have \( 2^1 = 2 \) elements: \( B = \{\emptyset, \{\emptyset\}\} \). The intersection \( A \cap B \) consists of elements common to both A and B. Both sets contain \( \emptyset \), so \( A \cap B = \{\emptyset\} \). It is important to distinguish the empty set \( \emptyset \) from the set containing the empty set \( \{\emptyset\} \).

🎯 Exam Tip: Be very careful with the notation \( \emptyset \) (empty set) and \( \{\emptyset\} \) (a set containing the empty set as an element). They are not the same! The empty set \( \emptyset \) has 0 elements, while \( \{\emptyset\} \) has 1 element (which is the empty set itself).

Question 12. In a class of 50 boys, 35 boys play carrom and 20 boys play chess then the number of boys play both games is........
(a) 5
(b) 30
(c) 15
(d) 10
Answer: (a) 5
In simple words: Let C be the set of boys who play carrom and S be the set of boys who play chess. We are given \( n(C \cup S) = 50 \) (total boys), \( n(C) = 35 \), and \( n(S) = 20 \). We use the formula for the number of elements in the union of two sets: \( n(C \cup S) = n(C) + n(S) - n(C \cap S) \). Plugging in the values, we get \( 50 = 35 + 20 - n(C \cap S) \). So, \( 50 = 55 - n(C \cap S) \), which means \( n(C \cap S) = 55 - 50 = 5 \). This tells us that 5 boys play both games.

🎯 Exam Tip: This is a standard problem using the principle of inclusion-exclusion. Always remember the formula \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \) to find the intersection or union when other values are given.

Question 13. If \( U = \{x : x \in N \text{ and } x < 10\}, A = \{1, 2, 3, 5, 8\} \text{ and } B = \{2, 5, 6, 7, 9\}, \text{ then } n[(A \cup B)'] \) is
(a) 1
(b) 2
(c) 4
(d) 8
Answer: (a) 1
In simple words: First, list the universal set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Then find the union of A and B: \( A \cup B = \{1, 2, 3, 5, 6, 7, 8, 9\} \). The complement of \( (A \cup B) \), denoted as \( (A \cup B)' \), includes all elements in U that are not in \( A \cup B \). Comparing U and \( A \cup B \), the only element in U but not in \( A \cup B \) is 4. So, \( (A \cup B)' = \{4\} \). The number of elements in this set, \( n[(A \cup B)'] \), is 1.

🎯 Exam Tip: Always list all elements of the sets clearly, especially for U, A, and B. This makes finding the union and then the complement much easier and reduces errors.

Question 14. For any three sets P, Q and R, \( P - (Q \cap R) \) is .........
(a) \( P - (Q \cup R) \)
(b) \( (P \cap Q) - R \)
(c) \( (P - Q) \cup (P - R) \)
(d) \( (P - Q) \cap (P - R) \)
Answer: (c) \( (P - Q) \cup (P - R) \)
In simple words: This is a distributive property for set difference over intersection, similar to how \( A - (B \cap C) = (A - B) \cup (A - C) \). It means the elements that are in P but not in both Q and R are the same as the elements that are in P but not in Q, combined with the elements that are in P but not in R. This identity helps simplify set expressions.

🎯 Exam Tip: Remember De Morgan's Laws and other set identities. \( A - B \) can be written as \( A \cap B' \). Using this, \( P - (Q \cap R) = P \cap (Q \cap R)' = P \cap (Q' \cup R') = (P \cap Q') \cup (P \cap R') = (P - Q) \cup (P - R) \). This method confirms the identity.

Question 15. Which of the following is true?
(a) \( A - B = A \cap B \)
(b) \( A - B = B - A \)
(c) \( (A \cup B)' = A' \cup B' \)
(d) \( (A \cap B)' = A' \cup B' \)
Answer: (d) \( (A \cap B)' = A' \cup B' \)
In simple words: Option (d) is one of De Morgan's Laws, which correctly states that the complement of the intersection of two sets is equal to the union of their complements. The other options are incorrect set identities. For example, \( (A \cup B)' = A' \cap B' \) (not \( A' \cup B' \)), and \( A - B \) is usually not equal to \( B - A \).

🎯 Exam Tip: De Morgan's Laws are fundamental: \( (A \cup B)' = A' \cap B' \) and \( (A \cap B)' = A' \cup B' \). Memorize these or be able to derive them using Venn diagrams to avoid common errors.

Question 16. If \( n(A \cup B \cup C) = 100, n(A) = 4x, n(B) = 6x, n(C) = 5x, n(A \cap B) = 20, n(B \cap C) = 15 \) and \( n(A \cap C) = 25 \), \( n(A \cap B \cap C) = 10 \), then the value of \( x \) is..........
(a) 10
(b) 15
(c) 25
(d) 30
Answer: (a) 10
In simple words: We use the formula for the number of elements in the union of three sets: \( n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \) Substitute the given values into the formula: \( 100 = 4x + 6x + 5x - 20 - 15 - 25 + 10 \) Combine the terms with \( x \): \( 100 = 15x - (20 + 15 + 25) + 10 \) \( 100 = 15x - 60 + 10 \) \( 100 = 15x - 50 \) Add 50 to both sides: \( 100 + 50 = 15x \) \( 150 = 15x \) Now, divide by 15 to find \( x \): \( x = \frac{150}{15} = 10 \). Thus, the value of x is 10.

🎯 Exam Tip: Ensure you remember the full inclusion-exclusion principle for three sets. Pay careful attention to signs (+/-) when substituting values and performing calculations, especially with the intersection terms.

Question 17. For any three sets A, B and C, \( (A - B) \cap (B - C) \) is equal to
(a) A only
(b) B only
(c) C only
(d) \( \emptyset \)
Answer: (d) \( \emptyset \)
In simple words: The set \( (A - B) \) contains elements that are in A but not in B. The set \( (B - C) \) contains elements that are in B but not in C. If we look for common elements between \( (A - B) \) and \( (B - C) \), we are looking for elements that are (in A and not in B) AND (in B and not in C). An element cannot be "not in B" and "in B" at the same time. Therefore, there are no common elements between these two sets, meaning their intersection is the empty set \( \emptyset \).

🎯 Exam Tip: When evaluating intersections of sets involving differences, it is often helpful to convert \( (X - Y) \) to \( X \cap Y' \). So \( (A - B) \cap (B - C) = (A \cap B') \cap (B \cap C') \). Since \( B' \cap B = \emptyset \), the entire expression becomes \( \emptyset \). This is a quick way to confirm the answer.

\( A - B \)

\( B - C \)

\( (A - B) \cap (B - C) = \{\} \)

Question 18. J = Set of three sided shapes, K = Set of shapes with two equal sides and L = Set of shapes with right angle, then \( J \cap K \cap L \) is.........
(a) Set of isosceles triangles
(b) Set of equilateral triangles
(c) Set of isosceles right triangles
(d) Set of right angled triangles
Answer: (c) Set of isosceles right triangles
In simple words: J describes triangles (three-sided shapes). K describes isosceles shapes (two equal sides). L describes right-angled shapes (one right angle). When we take the intersection \( J \cap K \cap L \), we are looking for shapes that fit all three descriptions: they must be triangles, have two equal sides, and contain a right angle. This perfectly defines an isosceles right triangle.

🎯 Exam Tip: Break down the definition of each set. Then, combine these definitions for the intersection. "Three-sided shapes" is another way of saying "triangles".

Question 19. The shaded region in the Venn diagram is

(a) \( Z - (X \cup Y) \)
(b) \( (X \cup Y) \cap Z \)
(c) \( Z - (X \cap Y) \)
(d) \( Z \cup (X \cap Y) \)
Answer: (a) \( Z - (X \cup Y) \)
In simple words: The shaded area in the Venn diagram shows the part of set Z that does not overlap with either set X or set Y. This represents all elements that are exclusively in Z, and not shared with any part of X or Y. This is precisely the definition of \( Z - (X \cup Y) \). It means Z *minus* the entire region covered by X or Y.

🎯 Exam Tip: When identifying shaded regions in Venn diagrams, start by understanding what each basic operation (union, intersection, difference, complement) looks like. Then, combine these operations to describe the shaded area accurately. \( A - B \) means "A only", so \( Z - (X \cup Y) \) means "Z only" (not touching X or Y).

Question 20. In a city, 40% people like only one fruit, 35% people like only two fruits, 20% people like all the three fruits. How many percentage of people do not like any three fruits?
(a) 5
(b) 8
(c) 10
(d) 15
Answer: (a) 5
In simple words: To find the percentage of people who do not like any of the fruits, we first calculate the total percentage of people who *do* like at least one fruit. We add the percentages for people who like only one fruit, only two fruits, and all three fruits: \( 40\% + 35\% + 20\% = 95\% \). Since the total population is 100%, the percentage of people who do not like any of the fruits is \( 100\% - 95\% = 5\% \).

🎯 Exam Tip: In percentage-based problems, assume a total of 100% for the entire group unless a specific number is given. This helps to easily calculate proportions for different categories. Make sure to identify if categories are mutually exclusive (like "only one fruit").