Venn Diagrams
A Venn diagram is a visual way to show the relationship between two or more sets. This page covers two-set Venn diagrams: how to draw them, how to place elements in the correct region, and how to read off intersections, unions, and complements. Worked examples and auto-marked practice questions are included.
What is a Venn Diagram?
A Venn diagram is a picture that shows how sets overlap. The rectangle stands for the universal set \(\xi\), which contains everything being considered in the problem. Each set is drawn as a circle inside the rectangle. Where the circles overlap, you have elements that are in both sets at the same time.
For two sets \(A\) and \(B\), the diagram has four regions:
The Words Behind the Symbols
Three small symbols do most of the work in Venn diagrams. Each one matches an everyday English word.
| Symbol | In English | What it means on a Venn diagram |
|---|---|---|
| \(\cap\) | and | Intersection. Elements that are in both sets at the same time. The overlap of the two circles. |
| \(\cup\) | or | Union. Elements that are in either set, or in both. Everything inside either circle. |
| \(A'\) | not | Complement. Elements that are not in \(A\). Everything outside circle \(A\), but still inside \(\xi\). |
So \(A \cap B\) reads as "A and B", \(A \cup B\) reads as "A or B", and \(A'\) reads as "not A". Watch out: \(\cup\) is "or" in the inclusive sense, so it includes elements that are in both sets.
The Four Regions
Elements in \(A\) but not in \(B\). The left part of circle A that does not overlap.
Elements in \(B\) but not in \(A\). The right part of circle B that does not overlap.
Elements in \(A\) and \(B\) at the same time. The lens-shaped overlap in the middle.
Elements inside \(\xi\) but outside both circles. They belong to the universal set, but not to \(A\) or \(B\).
Placing Elements into a Venn Diagram
To put elements into the correct region, work through the lists in this order:
- Find the intersection first. Any element that appears in both \(A\) and \(B\) goes in the overlap.
- Place the remaining elements of \(A\) in the A only region.
- Place the remaining elements of \(B\) in the B only region.
- Any element of \(\xi\) that is not in \(A\) or \(B\) goes outside both circles, but still inside the rectangle.
Worked example. Let \(\xi = \{1, 2, 3, \dots, 15\}\), \(A = \{\text{factors of } 12\}\) and \(B = \{\text{multiples of } 3 \text{ up to } 15\}\). Place every element in the correct region.
Listing each set: \(A = \{1, 2, 3, 4, 6, 12\}\) and \(B = \{3, 6, 9, 12, 15\}\). The intersection is \(A \cap B = \{3, 6, 12\}\), the values that appear in both lists.
Notice that every element of \(\xi\) appears exactly once on the diagram. If you find an element appearing twice, or you have an element left over, you have made a mistake in your sets.
Reading Information from a Venn Diagram
Once the diagram is complete, you can read off any set by collecting elements from the right regions.
A book club has 14 members. \(M\) is the set of members who enjoy mystery novels and \(S\) is the set of members who enjoy science fiction. The diagram below has been filled in with the number of people in each region.
From this diagram you can read off:
- \(n(M) = 4 + 3 = 7\): everyone inside circle \(M\), including the overlap.
- \(n(S) = 5 + 3 = 8\): everyone inside circle \(S\), including the overlap.
- \(n(M \cap S) = 3\): the overlap only.
- \(n(M \cup S) = 4 + 3 + 5 = 12\): everyone inside either circle, counting the overlap once.
- \(n(M') = 5 + 2 = 7\): everyone outside circle \(M\), inside or outside \(S\).
Watch out: the overlap of 3 belongs to both \(M\) and \(S\), so it is counted in \(n(M)\) and in \(n(S)\). When you compute \(n(M \cup S)\), you only add it in once.
Worked Examples
💡 Example 1: Counting Regions
A music school has 20 students. \(P\) is the set of students who play piano and \(V\) is the set of students who play violin. The Venn diagram is shown.
Find: (a) \(n(P)\), (b) \(n(P \cup V)\), (c) \(n(V')\).
(a) \(n(P) = 8 + 4 = 12\)
(b) \(n(P \cup V) = 8 + 4 + 6 = 18\)
(c) \(n(V') = 8 + 2 = 10\)
What's happening?
(a) Inside circle \(P\): the "P only" region plus the overlap.
(b) Everyone in \(P\), \(V\), or both. The overlap is added once.
(c) Outside circle \(V\): "P only" plus the "neither" region.
💡 Example 2: Placing Elements
\(\xi = \{x : x \text{ is an integer}, 1 \le x \le 10\}\), \(A = \{\text{even numbers}\}\) and \(B = \{\text{numbers greater than } 5\}\). Draw the Venn diagram with every element placed.
List each set:
\(A = \{2, 4, 6, 8, 10\}\)
\(B = \{6, 7, 8, 9, 10\}\)
Intersection (in both):
\(A \cap B = \{6, 8, 10\}\)
A only: \(\{2, 4\}\). B only: \(\{7, 9\}\). Neither: \(\{1, 3, 5\}\).
What's happening?
Find the overlap first by spotting numbers that appear in both lists.
Whatever is left in \(A\) goes in "A only". Whatever is left in \(B\) goes in "B only".
Anything in \(\xi\) not used yet goes outside both circles.
💡 Example 3: Filling in a Venn Diagram from Totals
In a tutor group of 30 students, 18 study French and 14 study Spanish. 6 students study both languages. Show the information on a Venn diagram and find how many students study neither French nor Spanish.
Start with the overlap (always work from the centre out):
\(n(F \cap S) = 6\).
French only: \(18 - 6 = 12\).
Spanish only: \(14 - 6 = 8\).
Total studying at least one language:
\(12 + 6 + 8 = 26\).
Neither subject:
\(30 - 26 = 4\).
So 4 students study neither.
What's happening?
The 18 who study French includes the 6 who study both. To find "French only" you subtract the overlap from the total who study French. The same idea works for "Spanish only".
The "neither" region is whatever is left over from the 30 once everyone in a circle is accounted for.
🔑 Key Points
- The rectangle is \(\xi\), the universal set. Sets are circles inside it.
- The overlap of two circles is \(A \cap B\), the elements in both sets.
- Everything inside either circle is \(A \cup B\).
- Everything outside circle \(A\) is \(A'\), the complement.
- Every element of \(\xi\) appears in exactly one region.
⚠️ Common Pitfalls
- Counting the overlap twice when working out \(n(A \cup B)\).
- Forgetting that the "neither" region is still part of \(\xi\).
- Writing \(n(A)\) as just the size of the "A only" region and missing the overlap.
- Confusing \(\cap\) with \(\cup\). Intersection is "and"; union is "or".
Looking Ahead: Three-Set Venn Diagrams
A two-set diagram has four regions. A three-set diagram (three overlapping circles) has eight regions, including a central region for elements in all three sets at once. The same skill, "work from the centre outwards", is the key technique for the harder questions you will meet in Sets 2. The next page covers exactly this.