Sets And Venn Diagrams
Master sets and Venn diagrams for Queensland Year 11 Mathematical Methods (QCAA). A sample space is a set of outcomes, and events are sets within it, giving clear language for describing events.
You will learn set notation for the complement of an event, the intersection and union of two events, recognise mutually exclusive events, and use Venn diagrams to illustrate these descriptions and region counts — the foundation of QCAA probability.
Every question with a fully worked solution.
- Sets And Venn Diagrams - Video - Venn Diagrams Watch
Theory
In Year 11 Mathematical Methods (QCAA, Unit 1 Topic 5), a set is a collection of outcomes and a Venn diagram pictures how events overlap. This page covers set notation \(A\cap B\), \(A\cup B\), \(A'\), reading region counts, the complement and De Morgan's laws, and the counting addition rule \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\).
A set is a collection of distinct objects called elements. The universal set \(\xi\) (or \(U\)) holds every outcome under consideration. We write \(x\in A\) for ‘\(x\) is in \(A\)’ and \(x\notin A\) for ‘\(x\) is not in \(A\)’.
The intersection \(A\cap B\) is the elements in both \(A\) and \(B\); the union \(A\cup B\) is the elements in either set (or both). The complement \(A'\) is everything in \(\xi\) that is not in \(A\). Two events are mutually exclusive when they cannot both happen, so \(A\cap B=\varnothing\).
De Morgan's laws connect complements to unions and intersections: \((A\cup B)'=A'\cap B'\) and \((A\cap B)'=A'\cup B'\).
The counting addition rule for the size of a union:
For mutually exclusive events \(A\cap B=\varnothing\), so the overlap is zero:
De Morgan's laws for complements:
How to solve a two-set Venn problem
- Draw two overlapping circles inside a rectangle for \(\xi\).
- Fill the overlap first: write \(n(A\cap B)\) in the middle, then subtract it to get \(A\) only \(=n(A)-n(A\cap B)\) and \(B\) only \(=n(B)-n(A\cap B)\).
- Find ‘neither’ by taking the three inner counts from \(n(\xi)\).
- Answer the question by reading or adding the required regions; for a probability, divide the favourable count by \(n(\xi)\).
List each set from the universal set:
| \(A\) | \(=\) | \(\{2,4,6,8,10\}\) |
| \(B\) | \(=\) | \(\{3,6,9\}\) |
Intersection — elements in both:
| \(A\cap B\) | \(=\) | \(\{6\}\) |
Union — elements in either (list once):
| \(A\cup B\) | \(=\) | \(\{2,3,4,6,8,9,10\}\) |
Complement — everything in \(\xi\) not in \(A\):
| \(A'\) | \(=\) | \(\{1,3,5,7,9\}\) |
\(A\cap B=\{6\}\), \(A\cup B=\{2,3,4,6,8,9,10\}\), \(A'=\{1,3,5,7,9\}\).
At least one — apply the addition rule:
| \(n(S\cup T)\) | \(=\) | \(n(S)+n(T)-n(S\cap T)\) |
| \(=\) | \(18+14-6\) | |
| \(=\) | \(26\) |
Neither — subtract from the whole class:
| \(n(\text{neither})\) | \(=\) | \(30-26\) |
| \(=\) | \(4\) |
\(26\) students play at least one sport; \(4\) play neither.
\(A\) only \(=A\cap B'\) — remove the overlap:
| \(n(A\cap B')\) | \(=\) | \(n(A)-n(A\cap B)\) |
| \(=\) | \(22-9\) | |
| \(=\) | \(13\) |
Union first, for the outside region:
| \(n(A\cup B)\) | \(=\) | \(22+17-9\) |
| \(=\) | \(30\) |
Outside both \(=(A\cup B)'=A'\cap B'\) by De Morgan:
| \(n(A'\cap B')\) | \(=\) | \(n(\xi)-n(A\cup B)\) |
| \(=\) | \(40-30\) | |
| \(=\) | \(10\) |
\(n(A\cap B')=13\) and \(n(A'\cap B')=10\).
Exactly one — the three ‘only’ regions:
| \(n(\text{exactly one})\) | \(=\) | \(8+6+5\) |
| \(=\) | \(19\) |
Exactly two — the three pairwise overlaps (not the centre):
| \(n(\text{exactly two})\) | \(=\) | \(4+3+2\) |
| \(=\) | \(9\) |
At least one — total inside the circles:
| \(n(\text{at least one})\) | \(=\) | \(8+6+5+4+3+2+1\) |
| \(=\) | \(29\) |
Probability \(=\dfrac{\text{favourable}}{\text{total}}\):
| \(P(\text{at least one})\) | \(=\) | \(\dfrac{29}{50}\) |
Exactly one: \(19\); exactly two: \(9\); \(P(\text{at least one})=\dfrac{29}{50}\).
Common pitfalls
Frequently asked questions
What is the difference between union and intersection?
\(A\cup B\) (union) is everything in either set; \(A\cap B\) (intersection) is only what is in both sets at once.
What does the addition rule for sets say?
\(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). You subtract the overlap once because it is counted in both \(n(A)\) and \(n(B)\).
What are mutually exclusive events?
Events that cannot both occur, so \(A\cap B=\varnothing\). Then \(n(A\cup B)=n(A)+n(B)\) with nothing to subtract.
What are De Morgan's laws?
\((A\cup B)'=A'\cap B'\) and \((A\cap B)'=A'\cup B'\). The complement of a union is the intersection of the complements.
How do you find the number in neither set?
Take the union from the universal set: \(n(A'\cap B')=n(\xi)-n(A\cup B)\).
How do you get a probability from a Venn diagram?
Divide the count in the favourable region by the total number of outcomes \(n(\xi)\).