Hard Level Worksheet
Very Short Answer Questions (1 Mark Each)
(1) Write the number of subsets of a set with 5 elements. Number of subsets =
Perfect! Using the formula
(2) If A = {x : x ∈ N and x < 4}, write A in roster form. A = {
Correct! Natural numbers less than 4 are 1, 2, and 3.
(3) What is the complement of the universal set? =
Excellent! The complement of the universal set is the empty set.
(4) Define a finite set with an example. A finite set has a
Perfect! Finite sets have a definite, countable number of elements.
(5) What is the power set of the empty set? P(∅) = {
Correct! The power set of empty set contains only the empty set itself.
Short Answer Questions (2 Marks Each)
Note: Answer each question with steps and explanation, in 2-3 sentences. Write down the answers on sheet and submit to the school subject teacher.
(1) If A = {1, 2, 3, 4}, B = {2, 4, 6}, and U = {1, 2, 3, 4, 5, 6}, find (i) A′ and (ii) B′.
A′ = U - A = {
B′ = U - B = {
Excellent! Complement contains all elements in U that are not in the given set.
(2) Let A = {a, b, c}, B = {b, c, d}. Find (i) A ∩ B, (ii) A ∪ B, and represent the results using a Venn diagram.
A ∩ B = {
A ∪ B = {
Perfect! Intersection finds common elements, union combines all elements.
(3) If A = {2, 4, 6}, B = {3, 6, 9}, and C = {6}, verify: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
B ∪ C = {
A ∩ (B ∪ C) = {
A ∩ B = {
(A ∩ B) ∪ (A ∩ C) = {
Excellent verification! This proves the distributive property.
(4) Draw a Venn diagram for three sets A, B, and C, such that A ∩ B = ∅, B ⊂ C.
A and B are
B is a
Perfect understanding! This creates specific set relationships in the diagram.
(5) Let U = {1, 2, 3, ..., 10}, A = {2, 4, 6, 8, 10}, and B = {1, 2, 3, 4, 5}. Find (i) A ∪ B, (ii) A ∩ B, (iii) A′.
A ∪ B = {
A ∩ B = {
A′ = {
Excellent comprehensive set operations!
Long Answer Questions (4 Marks Each)
Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.
(1) Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {1, 3, 5, 7}. Verify: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
B ∪ C = {
A ∩ (B ∪ C) = {
A ∩ B = {
A ∩ C = {
(A ∩ B) ∪ (A ∩ C) = {
Perfect verification! Both sides are equal, proving the distributive law.
(2) Prove that the number of elements in A ∪ B is given by: n(A ∪ B) = n(A) + n(B) – n(A ∩ B). Using A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6}, find n(A ∪ B).
n(A) =
A ∩ B = {
A ∪ B = {
Verification: n(A) + n(B) - n(A ∩ B) =
Excellent! This proves the inclusion-exclusion principle.
(3) A survey shows: 45 students like Mathematics, 30 like Science and 15 like both. Find how many students like only one subject and represent this using a Venn diagram.
Only Mathematics =
Only Science =
Students liking only one subject =
Total students surveyed =
Perfect application of set theory to real-world problems!
(4) Let A = {x : x is a factor of 12}, B = {x : x is a factor of 18}. Find A ∪ B and A ∩ B.
A = {
B = {
A ∪ B = {
A ∩ B = {
Excellent! Common factors are the common divisors of both numbers.
(5) In a class of 60 students, 36 like English, 27 like Telugu, and 12 like both. Find the number of students who like: (i) Only English (ii) Only Telugu (iii) Neither.
Only English =
Only Telugu =
Students liking at least one =
Students liking neither =
Perfect systematic analysis using Venn diagram principles!
Part B: Objective Questions (1 Mark Each)
Choose the correct answer and write the option (a/b/c/d)
(1) If A = {1, 3, 5}, B = {2, 4, 6}, then A ∩ B =
(a) {1, 2, 3, 4, 5, 6} (b) {} (c) {1, 3, 5} (d) {2, 4, 6}
Correct! These sets have no common elements, so their intersection is empty.
(2) If a set has 3 elements, the number of elements in its power set is
(a) 6 (b) 8 (c) 4 (d) 9
Correct! For n elements, power set has
(3) The universal set for A = {2, 4}, B = {4, 6} is
(a) {2, 4} (b) {2, 4, 6} (c) {4, 6} (d) {2, 6}
Correct! Universal set must contain all elements from both sets A and B.
(4) If A ∩ B = A, then
(a) A ⊂ B (b) B ⊂ A (c) A = B (d) A ∪ B = ∅
Correct! If intersection equals A, then all elements of A are in B, so A ⊂ B.
(5) The number of subsets of a set with 6 elements is
(a) 64 (b) 36 (c) 32 (d) 128
Correct! Using
(6) If A = {a, b}, then the power set of A is
(a) {a, b, ab} (b) {{}, {a}, {b}, {a, b}} (c) {{a, b}} (d) {ab, a, b, ∅}
Correct! Power set contains all possible subsets: empty set, singleton sets, and the full set.
(7) If A = {1, 2}, B = {2, 3}, then A′ ∩ B′ =
(a) {1} (b) {2} (c) {3} (d) Cannot be determined without U
Correct! Complements depend on the universal set, which is not given.
(8) Which of the following identities is true for any sets A and B?
(a) A ∪ A = ∅ (b) A ∩ A = ∅ (c) A ∪ A = A (d) A ∪ ∅ = ∅
Correct! This is the idempotent law: union of a set with itself equals the set.
(9) In set theory, the symbol ∅ denotes
(a) Universal set (b) Complement (c) Empty set (d) Power set
Correct! ∅ (or φ) represents the empty set containing no elements.
(10) If A ⊂ B and B ⊂ C, then
(a) A ⊂ C (b) C ⊂ A (c) A = C (d) A ∩ C = ∅
Correct! This demonstrates the transitivity property of subset relations.
Sets Challenge
Determine whether these statements about sets are True or False:
Sets Quiz
🎉 You Did It! What You've Learned:
By completing this worksheet, you now have a solid understanding of:
(1) Advanced Set Operations: Union, intersection, complement, and their properties
(2) Power Sets: Understanding that a set with n elements has 2ⁿ subsets
(3) Set Relationships: Subsets, proper subsets, and set equality
(4) Venn Diagrams: Visual representation of complex set relationships
(5) Set Identities: Distributive, associative, and De Morgan's laws
(6) Real-world Applications: Using sets to solve survey and classification problems
(7) Inclusion-Exclusion Principle: Formula for counting elements in unions
(8) Advanced Problem Solving: Applying set theory to mathematical and practical scenarios
Excellent work mastering advanced set theory concepts and their applications!