CBSE Class 11 Mathematics Set Theory Assignment Set 01

Multiple Choice Questions

Question. If A ={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then insert appropriate symbol ∈or Ïin each of the following blank spaces.
(i) 4 ... A (ii) − 4 ... A (iii) 12 ... A are
(a) ∈, ∈, ∈
(b) ∈, ∉, ∈
(c) ∈, ∉, ∉
(d) ∉, ∉, ∉
Answer : C

Question. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 3, 5, 7}. Then, which of following is true?
(a) A ∩ B = A
(b) A ∩ B =B
(c) A ∩ B ⊄ B
(d) None of these
Answer : B

Question. The set A = {14, 21, 28, 35, 42, ..., 98} in set-builder form is
(a) A = {x :x = 7n,n ∈N and 1 ≤n ≤15}
(b) A = {x :x = 7n,n ∈N and 2 ≤n ≤14}
(c) A = {x :x = 7n,n ∈N and 3 ≤n ≤13}
(d) A = {x :x = 7n,n ∈N and 4 ≤n ≤12}
Answer : B

Question. The set of months of a year is …X… set. Here, X refers to
(a) empty
(b) finite
(c) infinite
(d) singleton
Answer : B

Question. The following set in Roster form is {x : x is positive integer and a divisor of 9}
(a) {1, 3, 9}
(b) {1, 3, 8}
(c) {9, 8, 27}
(d) None of these
Answer : A

Question. Let A = {x : x is a square of a natural number and x is less than 100} and B is a set of even natural numbers. The cardinality of A ∩ B is
(a) 4
(b) 5
(c) 9
(d) None of these
Answer : A

Question. If A = the set of letters in ‘ALLOY’ and B = the set of letters in ‘LOYAL’, then A and B are …X… . Here, X refers to
(a) equal
(b) unequal
(c) disjoint
(d) None of these
Answer : A

Question. If A = {2, 4, 6, 8} and B = {6,8,10,12}, then A ∪ B is
(a) {2, 4, 6, 8}
(b) {6, 8, 10, 12}
(c) {6, 8}
(d) {2, 4, 6, 8, 10, 12}
Answer : A

Question. If X and Y are two sets such that X has 40 elements, X υ Y has 60 elements and X ∩ Y has 10 elements, then the number of elements does Y have
(a) 10
(b) 20
(c) 30
(d) 40
Answer : C

Question. If X = {8ⁿ – 7n – 1| n ∈ N } and y = {49n – 49 |n∈N}. Then,
(a) X ⊂ Y
(b) Y ⊂ X
(c) X =Y
(d) X ∩ Y = f
Answer : A

Question. The set of all natural numbers x such that 4x + 9 < 50 in roster form is
(a) {1, 2, 4, 6, 8, 10}
(b) {1, 3, 5, 7, 9}
(c) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(d) None of the above
Answer : C

Question. Let X = {Ram, Geeta, Akbar} be the set of students of class XI, who are in school hockey team and Y = {Geeta, David, Ashok} be the set of students from class XI, who are in the school football team. Then, X ∩ Y is
(a) {Ram, Geeta}
(b) {Ram}
(c) {Geeta}
(d) None of these
Answer : C

Question. Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second set. The values of m and n are, respectively
(a) 4, 7
(b) 7, 4
(c) 4, 4
(d) 7, 7
Answer : B

Question. If a set is denoted as A = Φ, then number of elements in A is
(a) 0
(b) 1
(c) 2
(d) 3
Answer : A

Question. The set {1, 2, 3, ...} is …Y… set. Here, Y refers to
(a) null
(b) finite
(c) infinite
(d) singleton
Answer : C

SET THEORY

Question Q1. (i) Write the set \(A = \left\{ \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8} \right\}\) in the set builder form
Answer: \(A = \left\{ x : x = \frac{y}{y + 1}, y \in \mathbb{N}, 2 \leq y \leq 7 \right\}\)
In simple words: Each element is a fraction where the numerator is a natural number and the denominator is one more than the numerator.

Exam Tip: Identify the pattern relating the numerator and denominator to express the general term algebraically.

Question Q1. (ii) Write the set \(A = \{1, 4, 9, 16, 25, 36\}\) in the set builder form
Answer: \(A = \{x : x = y^2, y \in \mathbb{N}, 1 \leq y \leq 6\}\)
In simple words: The elements are squares of consecutive natural numbers from 1 to 6.

Exam Tip: Recognize standard sequences like squares, cubes, or primes to convert between forms easily.

Question Q2. Write the following in roaster form
(i) \(A = \{x : x \text{ is a prime number less than 10}\}\)
Answer: \(A = \{2, 3, 5, 7\}\)
In simple words: Prime numbers are numbers greater than 1 that have only two factors: 1 and themselves.

Exam Tip: Remember that 1 is neither prime nor composite.

Question Q2. (ii) \(A = \{x : x \text{ is an integer, } x^2 \leq 9\}\)
Answer: \(A = \{-3, -2, -1, 0, 1, 2, 3\}\)
In simple words: Find all integers whose square is less than or equal to 9.

Exam Tip: Do not forget negative integers when squaring to find inequalities.

Question Q3. Write all the subsets of \(\{a, b\}\). If a set has 'n' elements then find the number of subsets
Answer: Subsets of \(\{a, b\}\) are \(\{\}, \{a\}, \{b\}, \{a, b\}\). Number of subsets for a set with \(n\) elements is \(2^n\).
In simple words: Every element has two choices - either to be included in a subset or not, giving \(2^n\) total combinations.

Exam Tip: The empty set \(\emptyset\) is a subset of every set.

Question Q4. Let \(A = \{1, 3, 5, 7, 9, 11\}\), \(B = \{2, 4, 6, 8, 10\}\) Find \(A \cup B\); \(A \cap B\)
Answer:
\(A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\)
\(A \cap B = \emptyset\)
In simple words: Union combines all unique elements from both sets. Intersection finds common elements (none here, so it is an empty set).

Exam Tip: When sets have no common elements, their intersection is always the null set \(\emptyset\).

Question Q5. Let \(U = \{1, 2, 3, \dots, 10\}\), \(A = \{1, 2, 4, 5, 10\}\), \(B = \{3, 4, 9, 10\}\) Find \((A \cup B)^c\); \(A - B\); \((A \cap B)^c\)
Answer:
\(A \cup B = \{1, 2, 3, 4, 5, 9, 10\}\)
\((A \cup B)^c = \{6, 7, 8\}\)
\(A - B = \{1, 2, 5\}\)
\(A \cap B = \{4, 10\}\)
\((A \cap B)^c = \{1, 2, 3, 5, 6, 7, 8, 9\}\)
In simple words: Complement means elements in the universal set \(U\) that are not in the given set. Set difference \(A - B\) removes elements of \(B\) from \(A\).

Exam Tip: Always subtract the universal set elements carefully when finding complements.

Question Q6. Let \(U = \{1, 2, \dots, 6\}\); \(A = \{2, 3\}\); \(B = \{4, 5, 6\}\) Verify Demorgan's Law i.e. (i) \((A \cup B)^c = A^c \cap B^c\) (ii) \((A \cap B)^c = A^c \cup B^c\)
Answer:
Given \(U = \{1, 2, 3, 4, 5, 6\}\), \(A = \{2, 3\}\), \(B = \{4, 5, 6\}\)
\(A^c = \{1, 4, 5, 6\}\), \(B^c = \{1, 2, 3\}\)
(i) \(A \cup B = \{2, 3, 4, 5, 6\}\)
\((A \cup B)^c = \{1\}\)
\(A^c \cap B^c = \{1, 4, 5, 6\} \cap \{1, 2, 3\} = \{1\}\)
Hence, \((A \cup B)^c = A^c \cap B^c\) is verified.
(ii) \(A \cap B = \emptyset\)
\((A \cap B)^c = U = \{1, 2, 3, 4, 5, 6\}\)
\(A^c \cup B^c = \{1, 4, 5, 6\} \cup \{1, 2, 3\} = \{1, 2, 3, 4, 5, 6\}\)
Hence, \((A \cap B)^c = A^c \cup B^c\) is verified.
In simple words: De Morgan's laws show how complements distribute over unions and intersections, flipping the operation symbol.

Exam Tip: To verify set laws, compute LHS and RHS separately and show they result in identical sets.

Question Q7. Draw Venn diagram of the following (i) \((A \cup B)^c\) (ii) \((A \cap B)^c\)
Answer:
(i) For \((A \cup B)^c\), shade the region outside both sets \(A\) and \(B\) within the universal set rectangle.
(ii) For \((A \cap B)^c\), shade the entire universal set except the overlapping intersection region of \(A\) and \(B\).
In simple words: A Venn diagram visually represents sets as circles inside a universal rectangle, with shaded regions showing set operations.

Exam Tip: Clearly label the universal set \(U\) and all component sets in every Venn diagram.

Question Q8. Write down all possible proper subsets of a set \(\{a, b, c\}\). Also if a set has 'n' elements. Find the number of proper subsets
Answer: Proper subsets of \(\{a, b, c\}\) are: \(\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}\). Number of proper subsets for a set with \(n\) elements is \(2^n - 2\).
In simple words: A proper subset cannot be equal to the original set itself, and usually excludes the empty set depending on definition, but standard formula excludes the set itself and sometimes the empty set. Here total subsets minus the set itself gives \(2^n - 1\), or excluding both empty set and set itself gives \(2^n - 2\). Following standard textbook convention: \(2^n - 1\) or \(2^n - 2\). The answer key specifies \(2^n - 2\).

Exam Tip: Remember that a proper subset strictly excludes the set itself.

Question Q9. Let \(X = \{\emptyset, \{\emptyset\}, 7, \{1, \emptyset\}, 2\}\), which of the following statements are true
(i) \(\emptyset \in X\)          (iv) \(\{\emptyset\} \subset X\)
(ii) \(\{1, \emptyset\} \in X\)        (v) \(1 \in X\)
(iii) \(\{\emptyset\} \in X\)         (vi) \(\{7\} \subset X\)
Answer:
(i) T
(ii) T
(iii) T
(iv) F
(v) F
(vi) T
In simple words: Check if an item is listed directly as an element (using \(\in\)) or if it can be formed as a subset (using \(\subset\)).

Exam Tip: Pay close attention to nested brackets when determining set membership vs subset relations.

Question Q10. Indicate finite or In-finite sets
(i) \(N = \{1, 2, 3, \dots\}\)
(ii) \(A = \{x : x \in \mathbb{Z} \text{ and } x^2 + 5x + 6 = 0\}\)
(iii) \(B = \{\text{All prime nos are divisible by itself}\}\)
(iv) \(C = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}\)
Answer:
(i) Infinite
(ii) Finite
(iii) Infinite
(iv) Infinite
In simple words: Finite sets have a countable, limited number of elements. Infinite sets go on endlessly without end.

Question Q11. Write the following in set builder form
(i) \((-5, 5)\)          (iii) \([5, 10)\)
(ii) \([-3, 4]\)          (iv) \((-3, 7]\)
Answer:
(i) \(\{x : x \in \mathbb{R}, -5 < x < 5\}\)
(ii) \(\{x : x \in \mathbb{R}, -3 \leq x \leq 4\}\)
(iii) \(\{x : x \in \mathbb{R}, 5 \leq x < 10\}\)
(iv) \(\{x : x \in \mathbb{R}, -3 < x \leq 7\}\)
In simple words: Open brackets \(()\) mean strict inequalities (\(<, >\)), while closed brackets \([]\) mean inclusive inequalities (\(\leq, \geq\)).

Exam Tip: Always specify the universal domain (like \(x \in \mathbb{R}\)) when writing intervals in set-builder form.