Sets · step 6 of 13

Cardinality & inclusion–exclusion

Adding |A| and |B| double-counts everything in both, so |A ∪ B| = |A| + |B| − |A ∩ B|. For three sets you subtract the three pairwise overlaps and add the triple overlap back, because it was removed once too often. The alternating pattern continues for any number of sets, and survey-style word problems are almost always this formula in disguise.

Practise cardinality & inclusion–exclusion free → All Sets subtopics

✓ Unlimited questions · marked criterion by criterion · no card needed

Method: how to approach it

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

Name the sets from the wordingTurn "students taking French" into a set F with a stated cardinality. Ambiguity in the English is the main source of error, not the arithmetic.
Write the formula for the right number of setsTwo sets: |A|+|B|−|A∩B|. Three: |A|+|B|+|C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|.
Read "only" and "exactly" carefully"Only A" means |A| minus everything shared. These phrases refer to regions of the Venn diagram, not to the raw cardinalities.
Check the totalThe disjoint regions must sum to |A ∪ B ∪ C|, and adding those outside gives |U|. This check catches nearly every slip.

Worked example

In a class of 40, 22 study French, 17 study German and 8 study both. How many study neither?

|F ∪ G| = |F| + |G| − |F ∩ G| = 22 + 17 − 8.
= 31 students study at least one language.
Neither = |U| − |F ∪ G| = 40 − 31.
Sanity check by regions: only French 14, only German 9, both 8, neither 9 — total 40 ✓.

Answer. 9 students study neither language.

Where marks get dropped

These are the specific errors that cost credit on cardinality & inclusion–exclusion questions — QED's rubric penalises each of them separately.

Adding cardinalities without subtracting the overlap, which counts the shared elements twice.
Getting the sign pattern wrong for three sets. The triple intersection is ADDED back, not subtracted.
Reading "22 study French" as "22 study only French". Unless the question says "only", the figure includes those who also study German.

Practise this until it is automatic

Unlimited fresh questions

QED generates new cardinality & inclusion–exclusion problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.

Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

Every question you generate is kept and re-takeable as a timed exam, and your Sets mastery is tracked so you know when this is exam-ready.

Cardinality & inclusion–exclusion — frequently asked questions

What is the formula for four sets?

All singles, minus all pairs, plus all triples, minus the quadruple intersection — the signs alternate with the size of the intersection. In general the term for a k-fold intersection carries sign (−1)^(k+1).

When are the sets disjoint enough to just add?

When every pairwise intersection is empty. Then all correction terms vanish and |A ∪ B| = |A| + |B|, which is the sum rule.

Is a Venn diagram acceptable working?

Yes for counting problems — filling regions from the innermost outward is a standard and fully creditable method, and it is usually faster than the formula.

The rest of Sets

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

Ready to make cardinality & inclusion–exclusion exam-proof?

Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.

Start practising free →