Sets & Set Operations
Set terminology and notation, the number systems, equality, subsets and proper subsets, disjoint sets, Venn diagrams, union, intersection, complement and difference, set laws, De Morgan's laws, and inclusion–exclusion.
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Sets & Set Operations – Part I
Set terminology, notation, subsets and Venn diagrams.
00:03Lesson 2: sets and learning outcomes02:54Elements and the "not an element" notation03:47Roster (tabular) form: odd positive integers < 1004:41Set-builder notation06:30The empty (null) set07:26Number sets: N, Z, Q, R10:02Rational numbers: p/q with q ≠ 012:37Equal sets: order and repetition don't matter15:15Subsets17:05Proper subsets18:10Listing all subsets: 8 subsets of a 3-element set20:48Disjoint sets21:42Venn diagramsSets & Set Operations – Part II
Set operations, laws and counting elements.
00:45Set union02:38Set intersection03:38Complement05:27Set difference06:33Worked example: union, intersection, complement, difference09:28Complementation rules10:25Properties: commutative, associative, distributive laws14:54De Morgan's law worked with U = {1, …, 10}18:42Cars example: transmission, air conditioning, side airbags27:56Finite sets and cardinality29:40Inclusion–exclusion: n(A ∪ B)31:35Coffee drinkers: sugar and cream example1Set terminology & notation
| Definition | Meaning | Example |
|---|---|---|
| Set (2.1.1) | An unordered collection of objects | All students taking discrete maths at any school |
| Element / member (2.1.2) | An object in a set. a ∈ A: a is an element of A. a ∉ A: a is not | 3 ∈ {1, 3, 5}; 4 ∉ {1, 3, 5} |
| Universal set U (2.1.3) | The set of all elements under consideration | All living things on Earth |
| Empty (null) set | The set with no elements: ∅ or { } | The set of dogs with six legs |
Two ways to specify a set
1. Tabular (roster) form
List all the elements.
O = {1, 3, 5, 7, 9}
2. Set-builder notation
Give the property members must have.
O = {x | x is an odd positive integer less than 10}
Number sets you must know
| Symbol | Set | Definition |
|---|---|---|
| ℝ | Real numbers | {x : −∞ < x < ∞} |
| ℤ | Integers | {…, −3, −2, −1, 0, 1, 2, 3, …} |
| ℤ⁺ | Positive integers | {1, 2, 3, …} |
| ℤ⁻ | Negative integers | {−1, −2, −3, …} |
| ℕ₀ | Natural numbers with zero = whole numbers | {0, 1, 2, 3, …} |
| ℕ | Natural numbers | {1, 2, 3, …} |
| ℚ | Rational numbers | {x : x = p/q, p ∈ ℤ, q ∈ ℤ, q ≠ 0} |
Natural {1, 2, 3, …} ⊂ Whole (adds 0) ⊂ Integers (adds …, −2, −1) ⊂ Rational (adds 1/2, −5/9, 3/8, …) ⊂ Real
2Equality, subsets & disjoint sets
Equal sets (2.1.4)
A = B if and only if they have the same elements: ∀x (x ∈ A ⇔ x ∈ B).
- {1, 3, 5} = {3, 5, 1}. Order doesn't matter.
- {1, 3, 3, 3, 5, 5, 5, 5} = {1, 3, 5}. Repeats don't matter.
Subset (2.1.5) and proper subset (2.1.6)
| Concept | Notation | Meaning |
|---|---|---|
| Subset | A ⊆ B | Every element of A is also in B |
| Equality via subsets | A = B ⇔ A ⊆ B and B ⊆ A | The standard way to prove two sets are equal |
| Proper subset | A ⊂ B | A ⊆ B but A ≠ B (B has at least one extra element) |
| Not a subset | D ⊄ A | Some element of D isn't in A |
A = {1, 3, 5}
- B = {1, 5} → B ⊂ A (a proper subset)
- C = {3, 1, 5} → C ⊆ A, but not proper, since C = A
- D = {1, 4} → D ⊄ A, since 4 ∉ A
- 0 elements: ∅
- 1 element: {a}, {b}, {c}
- 2 elements: {a, b}, {a, c}, {b, c}
- 3 elements: {a, b, c}
That's 8 = 2³ subsets. In general, a set with n elements has 2ⁿ subsets (this collection is the power set).
- ∅ ⊆ A for every set A. But ∅ ∈ A only if ∅ is literally listed as an element.
- 0 ≠ ∅. 0 is a number; ∅ is a set.
- A ⊆ A is always true. A ⊂ A is always false.
- {a, b} ∈ {a, b, c} is false: {a, b} is a subset, not an element.
Disjoint sets (2.1.7)
A and B are disjoint if they have no elements in common: A ∩ B = ∅.
Example 2.1.7.1: A = {1, 2}, B = {4, 5, 6}, C = {5, 6, 7, 8}. A and B are disjoint and A and C are disjoint, but B and C are not (they share 5 and 6).
3Venn diagrams
Venn diagrams show sets visually:
- The universal set U is a rectangle.
- Subsets of U are regions inside it (usually circles).
| Picture | Relationship |
|---|---|
| One circle labelled "A, B" | A = B |
| Two separate circles | A and B are disjoint |
| Circle A inside circle B | A ⊂ B |
4Set operations
| Operation | Notation | Set-builder definition | Venn region |
|---|---|---|---|
| Union | A ∪ B | {x | x ∈ A or x ∈ B (or both)} | Both circles entirely |
| Intersection | A ∩ B | {x | x ∈ A and x ∈ B} | The overlap only |
| Complement | Aᶜ | {x | x ∈ U and x ∉ A} | Everything outside A |
| Difference | A \ B (or A − B) | {x | x ∈ A, x ∉ B} | A minus the overlap |
- A ∪ B = {a, b, c, d}
- A ∩ B = {a, c}
- Aᶜ = {d}
- A \ B = {b}
A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}. Then A ∩ B = ∅, so A and B are disjoint.
5Set laws
Complementation rules
a. Uᶜ = ∅ b. ∅ᶜ = U c. (Aᶜ)ᶜ = A d. A ∪ Aᶜ = U e. A ∩ Aᶜ = ∅
Properties of set operations
| Law | Union | Intersection |
|---|---|---|
| Commutative | A ∪ B = B ∪ A | A ∩ B = B ∩ A |
| Associative | A ∪ (B ∪ C) = (A ∪ B) ∪ C | A ∩ (B ∩ C) = (A ∩ B) ∩ C |
| Distributive | A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) | A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
De Morgan's laws
(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
Complementing a union or intersection flips ∪ ↔ ∩ and complements each set. It's the same pattern as ¬(p ∨ q) ≡ ¬p ∧ ¬q in logic. Check both sides in the shader: type (A ∪ B)' and then A' ∩ B'.
U = {1, …, 10}, A = {1, 2, 4, 8, 9}, B = {3, 4, 5, 6, 8}
Left side - A ∪ B = {1, 2, 3, 4, 5, 6, 8, 9} - (A ∪ B)ᶜ = {7, 10}
Right side - Aᶜ = {3, 5, 6, 7, 10} - Bᶜ = {1, 2, 7, 9, 10} - Aᶜ ∩ Bᶜ = {7, 10}
Both sides are equal. ✓
A = automatic transmission, B = air conditioning, C = side air bags.
| Description | Expression |
|---|---|
| (a) At least one option | A ∪ B ∪ C |
| (b) Exactly one option | (A ∩ Bᶜ ∩ Cᶜ) ∪ (B ∩ Cᶜ ∩ Aᶜ) ∪ (C ∩ Aᶜ ∩ Bᶜ) |
| (c) Automatic and side air bags but no air conditioning | A ∩ C ∩ Bᶜ |
The shader has buttons for (b) and (c).
6Number of elements — inclusion–exclusion
- If a set S has exactly n distinct elements (n a non-negative integer), S is finite and n is its cardinality, written |S| or n(S).
- Example: A = odd positive integers less than 10, so |A| = 5. Also |∅| = 0.
If A and B are disjoint: n(A ∪ B) = n(A) + n(B)
Why subtract? Elements in both A and B are counted twice when you add n(A) + n(B).
70 take sugar (A), 60 take cream (B), 50 take both.
| Question | Working | Answer |
|---|---|---|
| (a) Only sugar | n(A) − n(A ∩ B) = 70 − 50 | 20 |
| (b) Only cream | n(B) − n(A ∩ B) = 60 − 50 | 10 |
| (c) Sugar or cream | 70 + 60 − 50 | 80 |
| (d) Exactly one item | 20 + 10 | 30 |
(And 100 − 80 = 20 take neither.)
Expect:
- Writing sets in builder/tabular form
- True/false on ∈ vs ⊆ and ∅
- Listing all subsets
- Computing ∪ ∩ ᶜ \
- Verifying De Morgan by direct computation
- Writing set expressions for word problems (cars, students, hospital staff)
- Inclusion–exclusion survey problems: draw the Venn diagram and fill in the intersection first
7Quick revision sheet
| Concept | Remember it as |
|---|---|
| Set | Unordered collection; repeats and order ignored |
| ∈ / ⊆ / ⊂ | Element of / subset / proper subset (≠) |
| A = B | A ⊆ B and B ⊆ A |
| ∅ | Subset of every set; n(∅) = 0; ∅ ≠ 0 |
| Number of subsets | 2ⁿ |
| Number systems | ℕ ⊂ ℕ₀ ⊂ ℤ ⊂ ℚ ⊂ ℝ |
| A ∪ B / A ∩ B | or / and |
| Aᶜ / A \ B | Not in A / in A but not B |
| Disjoint | A ∩ B = ∅ |
| De Morgan | (A∪B)ᶜ = Aᶜ∩Bᶜ; (A∩B)ᶜ = Aᶜ∪Bᶜ |
| Laws | Commutative, associative, distributive |
| Inclusion–exclusion | n(A∪B) = n(A) + n(B) − n(A∩B) (+ 3-set version) |
?Self-check quiz
11 questions on this week. Answer them all, then press Check my answers. Your best and latest scores are saved and shown on the subject page.