Sets · step 11 of 13

Venn diagrams & shading regions

A Venn diagram with n sets divides the universe into 2ⁿ regions, one for each combination of in/out choices — four regions for two sets, eight for three. Every set expression corresponds to a union of some of these regions, so shading is really a question about which of the 2ⁿ membership patterns satisfy the expression. That reframing makes even nested expressions routine.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Label every regionFor three sets label the eight regions by which of A, B, C the elements belong to, including the outside region.
Test each region against the expressionTake a representative element of the region, evaluate the expression’s membership condition, and shade if it comes out true.
Build up from the insideFor nested expressions, shade the innermost bracket lightly first, then apply the outer operation to that shading.
Fill numbers innermost-first for countingStart with |A ∩ B ∩ C|, then work outward subtracting what is already placed. Each region should end with a non-negative number.

Worked example

Which region does A \ (B ∪ C) occupy, and how many of the eight three-set regions does (A ∩ B) ∪ C cover?

A \ (B ∪ C) requires in A, not in B, not in C — exactly one region.
For the second expression, list regions where the condition holds.
C alone, C∩A, C∩B, C∩A∩B — four regions from C.
Plus A∩B with C excluded — one more.

Answer. A \ (B ∪ C) is the single "A only" region; (A ∩ B) ∪ C covers five of the eight regions.

Where marks get dropped

These are the specific errors that cost credit on venn diagrams & shading regions questions — QED's rubric penalises each of them separately.

Forgetting the region outside all circles. It is a genuine region of the universe and is included in complements.
Shading A ∩ Bᶜ as though it were (A ∩ B)ᶜ. The complement’s scope changes the answer completely.
Trying to draw four sets with circles. Four circles cannot produce all 16 regions — ellipses or a different construction are required.

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Answer in real notation

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Venn diagrams & shading regions — frequently asked questions

Can a Venn diagram prove a set identity?

It is convincing and standard for two or three sets, but many courses require a formal element-chase or algebraic proof. Use the diagram to find the answer, then write the proof.

How many regions do n sets create?

2ⁿ, one for each in/out pattern. That is why diagrams become unusable past three or four sets and why indicator functions take over.

What is the difference between a Venn and an Euler diagram?

A Venn diagram always shows all 2ⁿ regions, even empty ones. An Euler diagram draws only the non-empty ones, so it encodes extra information about the sets.

The rest of Sets

Set-builder notation, operations, and set identities. Each subtopic below has its own method, worked example and mark-losing traps.

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