Discrete Mathematics
Discrete mathematics deals with distinct, separate values rather than continuous data. In Grade 6 you will learn to organise information using sets and Venn diagrams — essential tools for logical reasoning.
What You'll Learn
- Understand what a set is and use correct notation
- Identify elements, subsets, and the universal set
- Draw and interpret Venn diagrams with two sets
- Find the intersection, union, and complement of sets
- Use Venn diagrams to solve counting problems
- Apply set language to real-world classification tasks
IB Assessment Focus
Criterion A: Select and apply correct set notation and Venn diagram methods.
Criterion B: Discover patterns in set relationships and describe them as rules.
Criterion C: Use correct mathematical vocabulary (intersection, union, complement) and show clear diagrams.
Criterion D: Apply sets and Venn diagrams to organise and interpret real-world data (e.g., survey results, classifications).
Key Vocabulary
| Term | Definition | Example |
|---|---|---|
| Set | A collection of distinct objects, called elements | A = {2, 4, 6, 8} |
| Element (∈) | A member of a set | 4 ∈ A means "4 is in set A" |
| Universal set (U) | The set of ALL elements being considered | U = {1, 2, 3, … 10} |
| Empty set (∅) | A set with no elements | ∅ = { } |
| Subset (⊆) | A set whose elements all belong to another set | {2, 4} ⊆ {1, 2, 3, 4, 5} |
| Intersection (∩) | Elements in BOTH sets | A ∩ B |
| Union (∪) | Elements in EITHER or BOTH sets | A ∪ B |
| Complement (A′) | Elements NOT in set A but in the universal set | If U = {1…10}, A = {2,4,6,8,10}, then A′ = {1,3,5,7,9} |
| Venn diagram | A diagram using overlapping circles to show relationships between sets | Two overlapping circles inside a rectangle (U) |
Sets & Elements
A set is a well-defined collection of distinct objects. Learning the notation and terminology is the foundation for all set work.
Set Notation
Sets are written using curly braces { }. Each object inside is an element (or member).
- A = {1, 3, 5, 7, 9} — the set of odd numbers from 1 to 9
- B = {red, blue, green} — a set of colours
- Elements are not repeated in a set: {1, 2, 2, 3} should be written as {1, 2, 3}
- The order does not matter: {3, 1, 2} is the same set as {1, 2, 3}
Membership Notation
- ∈ means "is an element of" — 5 ∈ A means "5 belongs to set A"
- ∉ means "is NOT an element of" — 4 ∉ A means "4 does not belong to set A"
Types of Sets
Finite set: Has a countable number of elements. Example: A = {2, 4, 6} has 3 elements.
Empty set (null set): Written as ∅ or { }. It contains no elements at all. Example: the set of months with 32 days = ∅.
Universal set (U): Contains ALL elements under discussion. Every other set in the problem is a subset of U.
Subsets
Set A is a subset of set B (written A ⊆ B) if every element of A is also in B.
If B = {1, 2, 3, 4, 5}, then {1, 3} ⊆ B ✓
{1, 6} is NOT a subset of B because 6 ∉ B.
Special cases: The empty set ∅ is a subset of every set. Every set is a subset of itself.
Counting Elements
The number of elements in a set is called its cardinality, written n(A).
If A = {2, 4, 6, 8, 10}, then n(A) = 5.
If B = ∅, then n(B) = 0.
Venn Diagrams
Venn diagrams use overlapping circles inside a rectangle to show how sets relate to each other visually.
Drawing a Venn Diagram
- Draw a rectangle and label it U (universal set)
- Draw two overlapping circles inside the rectangle — label them A and B
- Place elements that belong to both A and B in the overlapping region
- Place elements that belong to only A in the left-only part of circle A
- Place elements that belong to only B in the right-only part of circle B
- Place elements that belong to neither A nor B outside both circles but inside the rectangle
Reading a Venn Diagram
A Venn diagram divides elements into four regions:
| Region | Description | Notation |
|---|---|---|
| Only in A | In A but NOT in B | A ∩ B′ |
| Both A and B | In the overlap | A ∩ B |
| Only in B | In B but NOT in A | A′ ∩ B |
| Neither | Outside both circles | A′ ∩ B′ (or (A ∪ B)′) |
Example: Building a Venn Diagram
Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}
Step 1 — Intersection: A ∩ B = {3, 4, 5} → place in the overlap.
Step 2 — Only A: {1, 2} → place in the A-only region.
Step 3 — Only B: {6, 7} → place in the B-only region.
Step 4 — Neither: {8, 9, 10} → place outside both circles.
Set Operations
The three key operations — intersection, union, and complement — let you combine, compare, and exclude elements from sets.
Intersection (∩)
A ∩ B = {3, 5} — only these appear in both sets.
Union (∪)
A ∪ B = {1, 2, 3, 4, 5, 7, 9} — list every element from both sets (no repeats).
Complement (A′)
A′ = {1, 3, 5, 7, 9} — all elements in U that are not in A.
Key Relationships
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B) — the addition principle
- If A ∩ B = ∅, then A and B are disjoint (no common elements)
- n(A′) = n(U) − n(A)
Problem Solving with Sets
Venn diagrams are powerful tools for solving real-world counting problems, especially when items belong to more than one category.
Survey Problems
A common type of question gives you totals for two groups and asks you to find the overlap, or the number in neither group.
- Draw a Venn diagram with two circles (one for each category)
- If given, fill in the intersection first
- Subtract the intersection from each group’s total to fill the "only" regions
- Add up all regions inside the circles; subtract from the total to find "neither"
Step 1: Both = 5 → place 5 in the overlap.
Step 2: Only football = 18 − 5 = 13. Only basketball = 12 − 5 = 7.
Step 3: Total in at least one sport = 13 + 5 + 7 = 25.
Step 4: Neither = 30 − 25 = 5 students play neither sport.
The Addition Principle
n(Football ∪ Basketball) = 18 + 12 − 5 = 25
Neither = 30 − 25 = 5
Classification Problems
Sets can also classify objects by properties. For example, sorting numbers into "Even" and "Greater than 5" helps identify which numbers have both properties, only one, or neither.
A = {2, 4, 6, 8, 10}, B = {6, 7, 8, 9, 10}
A ∩ B = {6, 8, 10} — even AND greater than 5
Only A = {2, 4} — even but NOT greater than 5
Only B = {7, 9} — greater than 5 but NOT even
Neither = {1, 3, 5} — odd AND 5 or less
Worked Examples
These examples show the step-by-step reasoning expected in your assessments. Notice how each step uses correct set notation.
A ∪ B = elements in EITHER or BOTH = {2, 4, 6, 8, 10, 12, 16}.
Note: 4 and 8 appear in both sets but are listed only once in the union.
A′ = elements in U but NOT in A = {1, 2, 4, 5, 7, 8, 10, 11}. So n(A′) = 8.
Check: n(A) + n(A′) = 4 + 8 = 12 = n(U) ✓
Step 2: Only maths = 15 − 4 = 11. Only science = 10 − 4 = 6.
Step 3: Total who like at least one = 11 + 4 + 6 = 21.
Step 4: Neither = 28 − 21 = 7 students.
Using the formula: n(M ∪ S) = 15 + 10 − 4 = 21. Neither = 28 − 21 = 7 ✓
A ∩ B = {2} — the only even prime. Place 2 in the overlap.
Only A: {4, 6, 8, 10} — even but not prime.
Only B: {3, 5, 7} — prime but not even.
Neither: {1, 9} — not even and not prime.
Check: 4 + 1 + 3 + 2 = 10 = n(U) ✓
Cat only = 22 − 6 = 16.
Dog only = 16 − 6 = 10.
At least one pet = 16 + 6 + 10 = 32.
Neither = 40 − 32 = 8 people own neither.
Check: 16 + 6 + 10 + 8 = 40 ✓
Practice Q&A
Attempt each question before revealing the model answer. Focus on using correct set notation.
P ∪ Q = elements in either or both = {1, 2, 3, 5, 7, 8, 9}. We list every element from both sets without repeating.
A′ = all elements in U not in A = {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14}. n(A′) = 12.
Check: 3 + 12 = 15 = n(U) ✓
Neither = 35 − 30 = 5 students study neither language.
Step 2 — Only A: {1, 2}. Place in the A-only region.
Step 3 — Only B: {5, 6}. Place in the B-only region.
Step 4 — Neither: {7, 8}. Place outside both circles but inside the rectangle.
Check: 2 + 2 + 2 + 2 = 8 = n(U) ✓
A ∩ B = {1, 2, 3, 6}. These are the common factors of 12 and 18. Notice the largest is 6, which is the HCF — connecting sets to number theory.
Neither = n(U) − n(A ∪ B) = 30 − 20 = 10.
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.