SETS
Below is a comprehensive note covering the topic “sets”with details on types of sets, algebra of sets, Venn (commonly mis‐spelled “vein”) diagrams and their applications At the end you’ll find five worked examples (with diagrams) similar to JAMB exam questions, followed by ten practice questions for further review.
1. Introduction to Sets
2. Types of Sets
Sets come in many forms. Some important types include:
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Finite and Infinite Sets:– A finite set has a limited number of elements (e.g., {2, 4, 6, 8}).– An infinite set has endlessly many elements (e.g., the set of natural numbers ℕ = {0, 1, 2, 3, …}).
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Equal Sets:Two sets are equal if they have exactly the same elements. (Order is irrelevant.)
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Subsets and Proper Subsets:– A set A is a subset of B (A ⊆ B) if every element of A is also in B.– A is a proper subset of B (A ⊂ B) if A ⊆ B but A ≠ B.
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Universal Set:The universal set (usually denoted U) is the “universe” of all objects under discussion.
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Empty (Null) Set:The set with no elements is denoted by ∅.
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Special Number Sets:Common examples include:ℕ (natural numbers),ℤ (integers),ℚ (rational numbers),ℝ (real numbers), andℂ (complex numbers).
(For an in‐depth review of set types see Math is Fun on Sets and Venn Diagrams (Click here for more details )
3. Algebra of Sets
The algebra of sets involves operations that combine or modify sets. The basic operations include:
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Union ( ∪ ):The union of A and B, A ∪ B, is the set of all elements that are in A or B (or both).Example: If A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}.
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Intersection ( ∩ ):The intersection, A ∩ B, is the set of all elements common to both A and B.Example: A ∩ B = {2}.
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Complement ( Aᶜ or A′ ):The complement of A (relative to U) is the set of all elements in U that are not in A.
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Difference ( Relative Complement, A – B ):The difference A – B is the set of elements in A that are not in B.
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Symmetric Difference ( Δ ):A Δ B = (A – B) ∪ (B – A).
These operations satisfy many useful laws:
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Commutative Laws:A ∪ B = B ∪ A, A ∩ B = B ∩ A
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Associative Laws:(A ∪ B) ∪ C = A ∪ (B ∪ C), (A ∩ B) ∩ C = A ∩ (B ∩ C)
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Distributive Laws:A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
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De Morgan’s Laws:(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
4. Venn Diagrams and Their Applications
A Venn diagram is a graphical tool that uses overlapping closed curves (usually circles) to represent sets and illustrate their relationships. Each circle shows a set; the overlapping areas represent intersections, while the non‐overlapping parts show the difference between sets.
Applications include:
- Mathematics: Visualizing unions, intersections, and complements; solving counting problems.
- Logic and Probability: Illustrating logical relations and calculating probabilities from overlapping events.
- Real Life: Organizing data (e.g., survey results, student subject combinations) to quickly see commonalities and differences.
(For further background, see Cuemath on Venn Diagrams Click here for more details and Wikipedia’s Venn diagram article Click here for more details )
5. Worked Examples (JAMB Exam Style)
Below are five worked examples that mirror the style of JAMB exam questions. (Simple diagrams are provided using basic sketches.)
Example 1: Two-Set Union and Intersection
- Find A ∪ B and A ∩ B.
Venn Diagram Sketch:
_______
Example 2: Students Taking Subjects
In a class of 40 students, 32 offer Mathematics, 24 offer Physics, and 4 offer neither.
- Find how many offer both Mathematics and Physics.
Venn Diagram Sketch:
____________
Example 3: Intersection of Even Numbers and Common Factors
Example 4: Multiples and Parity
- Find A ∩ B.
Venn Diagram Sketch (Two overlapping circles):
_______ ← A: {3, 6, 9, 12, 15, 18}
The overlapping region contains {3, 9, 15}.
Example 5: Three-Set Venn Diagram Problem
- Find the number of elements only in A.
6. JAMB Practice Questions on Sets
Below are ten practice questions you can use to test your understanding:
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Question 1:If A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5}, what is A ∩ B?A) {2, 4} B) {1, 3, 5} C) {2, 4, 6} D) {2, 4, 8}
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Question 2:Let U = {1, 2, …, 15} and A = {multiples of 3}. Find Aᶜ (the complement of A in U).A) {1, 2, 4, 5, 7, 8, 10, 11, 13, 14} B) {3, 6, 9, 12, 15} C) {1, 3, 5, …} D) {2, 4, 8, 10, 14}
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Question 3:If A = {x: x is an even number ≤ 10} and B = {x: x is a prime number ≤ 10}, what is A ∪ B?A) {2, 3, 4, 5, 6, 7, 8, 10} B) {2, 3, 5, 7} C) {2, 4, 6, 8, 10} D) {2, 3, 4, 6, 7, 8, 10}
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Question 4:Let A, B be sets with |A| = 12, |B| = 9 and |A ∩ B| = 4. Find |A ∪ B|.A) 17 B) 21 C) 16 D) 25
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Question 5:If A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}, then A – B equals:A) {1, 2} B) {6, 7} C) {1, 2, 6, 7} D) {3, 4, 5}
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Question 6:For sets A and B, which of the following is equivalent to (A ∪ B)ᶜ?A) Aᶜ ∪ Bᶜ B) Aᶜ ∩ Bᶜ C) (A ∩ B)ᶜ D) (A – B)ᶜ
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Question 7:If U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}, then what is Aᶜ?A) {1, 3, 5} B) {2, 4, 6} C) {1, 2, 3, 4, 5, 6} D) ∅
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Question 8:Let A = {x: x is a multiple of 2, 1 ≤ x ≤ 20} and B = {x: x is a multiple of 5, 1 ≤ x ≤ 20}. Find A ∩ B.A) {10, 20} B) {2, 4, 6} C) {5, 10, 15, 20} D) {2, 10, 20}
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Question 9:If A ⊆ B and B ⊆ C, then which of the following is always true?A) A ⊆ C B) C ⊆ A C) A = B D) C = B
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Question 10:If A = {x: x is a vowel in “education”} and B = {x: x is a letter in “computer”}, what is A ∩ B?A) {o} B) {u, o} C) {e, o} D) ∅
Answer Key:
- A; 2. A; 3. A; 4. B; 5. A; 6. B; 7. A; 8. A; 9. A; 10. A
This detailed note combines fundamental definitions, operations, diagrammatic representations, and practical examples—all of which are essential for understanding set theory in both academic and exam (JAMB) contexts.
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