Venn Diagrams

What is a Venn Diagram?

A Venn diagram is a picture that shows how sets overlap. The rectangle stands for the universal set \(\mathcal{E}\), 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:

Placing Elements into a Venn Diagram

To put elements into the correct region, work through the lists in this order:

1. Find the intersection first. Any element that appears in both \(A\) and \(B\) goes in the overlap.
2. Place the remaining elements of \(A\) in the A only region.
3. Place the remaining elements of \(B\) in the B only region.
4. Any element of \(\mathcal{E}\) that is not in \(A\) or \(B\) goes outside both circles, but still inside the rectangle.

Worked example. Let \(\mathcal{E} = \{1, 2, 3, \dots, 15\}\), \(A = \{\text{factors of } 12\}\) 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 \(\mathcal{E}\) 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:

• \(n(M) = 7\): everyone inside circle \(M\), so the 4 in "M only" plus the 3 in the overlap.
• \(n(S) = 8\): everyone inside circle \(S\), so the 5 in "S only" plus the 3 in the overlap.
• \(n(M \cap S) = 3\): the overlap only.
• \(n(M \cup S) = 12\): add 4, 3 and 5, everyone inside either circle with the overlap counted once.
• \(n(M') = 7\): everyone outside circle \(M\), so the 5 in "S only" plus the 2 outside both.

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.

Three-Set Venn Diagrams

Two sets give four regions. Three sets give eight, because every element is either in or out of each of the three circles. The extra one that catches people out is the region in the very middle, \(A \cap B \cap C\), where all three overlap at once.

Notice that "A and B" on the picture means A and B but not C. In set notation that region is \(A \cap B \cap C'\), while the whole lens where circles \(A\) and \(B\) cross, centre included, is \(A \cap B\). Exam questions rely on that difference, so read the wording carefully: a total like "18 take both A and B" almost always includes the students who take all three.

That is why three-set problems are filled in from the centre outwards:

1. Put the number in \(A \cap B \cap C\) first. It is usually stated directly.
2. Fill each pair region by subtracting the centre from the pair total: the "A and B only" region is \(n(A \cap B) - n(A \cap B \cap C)\).
3. Fill each "only" region by subtracting everything already written inside that circle from the circle's total.
4. Add all seven region values and subtract from \(n(\mathcal{E})\) to get the "none of them" region outside.

Shading a Region, and Naming a Shaded One

The other half of this topic runs in both directions. You may be asked to shade the region a piece of notation describes, or to look at a shaded diagram and write down the region in set notation.

To shade a region, take the expression apart and shade one set at a time, each in a different direction:

• For an intersection \(\cap\), the answer is only where the shadings cross.
• For a union \(\cup\), the answer is everything that got shaded either way.
• For a complement \('\), shade everything except that set, including the space outside the circles.

Going the other way, name what you can see. Ask two questions in order: is the shaded part inside one circle only, or does it cross more than one? And does it include the space outside the circles? If it does, a complement is involved somewhere.

The same region often has more than one correct name, and both are worth marks. The middle diagram above is \((A \cup B)'\), but it is equally \(A' \cap B'\): "not in A or B" and "not in A and not in B" describe the same space. The matching pair is \((A \cap B)' = A' \cup B'\). These two rules are De Morgan's laws, and shading a copy of the diagram both ways is how you check one.