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DM topic guide

Venn Diagrams

Reason about inclusion, overlap, exclusion and counts.

Guide page4 of 7

Fill regions from the centre outward — place the most certain facts first, then use totals to find what's left.

The method — step by step

1
Draw the circles

Two sets = two overlapping circles. Three sets = three circles, each pair overlapping.

2
Place universal facts

'All A are B' → A sits inside B. 'No A are B' → circles are completely separate.

3
Place existential facts

'Some A are B' → at least one person sits in the overlap region. Mark it, but don't invent a number.

4
Fill the centre region first

In three-set counting problems, always place the triple overlap (centre) before working on pairwise overlaps.

5
Subtract outward

Each pairwise overlap given includes the centre. Subtract the triple overlap to find the A-and-B-only region.

The key idea in action

Step 1

Patients in at least one circle = total − neither = 100 − 23 = 77

Step 2

Add the two circles separately → 62 + 51 = 113. That's more than 77, because people with BOTH were counted twice.

Step 3

The extra = those double-counted people → 113 − 77 = 36 have BOTH symptoms

Worked example

EXAMPLESymptoms X and Y — how many have both?

Given

A respiratory clinic reviewed 100 patients admitted over one week. Doctors recorded which patients presented with symptom X (a persistent cough) and which presented with symptom Y (shortness of breath).
62 patients had symptom X.
51 patients had symptom Y.
23 patients had neither symptom.

Question

How many patients had both symptom X and symptom Y?

Working

NEITHER23X ONLY26BOTH36Y ONLY15Symptom X (total 62)Symptom Y (total 51)

How many patients are in at least one circle? → 100 total minus 23 with neither

= 77

Count up all of X and all of Y separately → 62 + 51

= 113

But only 77 people are actually in the circles — 113 is more than 77 because people with BOTH were counted twice. The extra 36 are those double-counted people.

= 36 ← answer

X only: take the X total and remove the 36 who also have Y → 62 − 36

= 26

Y only: take the Y total and remove the 36 who also have X → 51 − 36

= 15

Check — all four groups must add back to 100 → 26 + 36 + 15 + 23

= 100 ✓
AnswerC — 36 patients have both X and Y

Rules to remember

Circle size is irrelevant

Unless numbers are given, a larger drawn circle does not represent a larger group.

'Some' guarantees a shared region

'Some A are B' means at least one person is in the overlap — but gives no count.

Work centre-outward for counts

In three-set problems: find the triple overlap first, then pairwise-only regions, then exclusive regions.

Double-subtraction trap

The overlap figure given usually includes the centre. Subtract the triple overlap to get the A-and-B-only count.

Rearrange the idea

Overlap = A total + B total − union. You can find any one of these if you know the other three.

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