UCAT Decision Making Venn Diagrams: 2-Set and 3-Set Worked Examples (2026)

At TheUKCATPeople, I am Dr Akash, and Venn diagram questions are the most visual part of Decision Making, and among the most learnable. Once you can place numbers into the right regions in the right order, they become quick, reliable marks. This guide gives you a clean method for two-set and three-set diagrams, shows you how to convert a table or a passage into a Venn, and draws every worked example so the numbers are in front of you.

In the 2026 UCAT, Decision Making is 35 questions in 37 minutes, just over a minute per question, with a simple on-screen calculator and no negative marking (UCAT Consortium).

The UCAT Consortium describes Decision Making by its answer formats rather than publishing a fixed list of named question types. The type names used across preparation, including this one, come from the official UCAT question tutorials and practice materials.

Venn diagram questions are single-answer multiple choice, so they are worth 1 mark each. The UCAT does not publish how many of the 35 Decision Making questions are Venn diagrams, so treat it as a handful rather than a fixed number.

This is the deep dive under the Decision Making complete guide. Drawing a Venn is also a powerful way to solve syllogisms, so the two guides reinforce each other.

The set language you need

Every Venn question is really a test of four phrases. Learn to map each to a region.

• Union, written A or B: everything inside either circle, counted once.
• Intersection, written A and B: only the overlap where both are true.
• Only A, or A but not B: the part of A outside the overlap.
• Neither: the region outside all the circles, inside the surrounding total.

In formal set notation you may see A ∪ B for the union, A ∩ B for the intersection, and n(A) for the number of items in set A. You do not need the symbols to answer the questions, but recognising them helps.

How to solve a two-set Venn

1. Draw two overlapping circles inside a rectangle. The rectangle is the whole group.
2. Fill the overlap first: the number that is in both sets.
3. Subtract the overlap from each circle total to get the two "only" regions.
4. Put any "in neither" figure outside both circles.
5. Reconcile: only A, plus only B, plus both, plus neither should equal the total. If not, you have mis-assigned a value.

Worked example 1: reading a two-set Venn

A clinic has 120 patients. The diagram shows how many have diabetes, hypertension, both or neither. How many patients have hypertension?

A) 22

B) 31

C) 53

D) 60

Working:

• Hypertension means the whole hypertension circle, not just the part outside diabetes.
• Whole circle = only hypertension + the overlap = 31 + 22 = 53.

Answer: C) 53

Trap B) 31 is only hypertension, and trap A) 22 is the overlap. Read the whole circle when the question names the set.

Converting text or a table into a Venn

Many questions give you a passage or a table and ask you to build the diagram yourself. Translate each sentence or cell into a region, always placing the overlap first.

Worked example 2: building a two-set Venn from text

In a study of 80 patients presenting to accident and emergency, 34 had chest pain, 27 had breathlessness, and 15 had both. How many patients had either chest pain or breathlessness, or both?

A) 46

B) 61

C) 15

D) 80

Working:

• Either or both is the union: |A| + |B| − |A and B|.
• 34 + 27 − 15 = 46.

Answer: A) 46

Trap B) 61 adds 34 and 27 but forgets to subtract the 15 counted in both circles. Always subtract the overlap once.

Worked example 3: converting a table into a Venn

The table shows 100 people by smoking status and whether they have a cough. How many are smokers who do not have a cough?

A) 12

B) 18

C) 27

D) 45

Working:

• Smokers who do not have a cough is the only-smoker region.
• All smokers = 45; smokers with a cough = 18; so only smokers = 45 − 18 = 27.

Answer: C) 27

Trap D) 45 is all smokers, and trap B) 18 is smokers who do have a cough.

How to solve a three-set Venn

Three circles create seven regions inside plus a "none" region outside. The reliable method is to work from the centre outward, because every overlap figure usually includes the ones nested inside it.

1. Place the triple overlap, in all three sets, at the very centre first.
2. Fill each pair-only region by subtracting the centre from that pairwise overlap.
3. Fill each "only" region by subtracting every overlap that touches it from the circle total.
4. Place the "none of the three" figure outside the circles.
5. Check that all eight regions add up to the total.

Worked example 4: a three-set Venn

200 medical students sat three exams. The diagram shows how many passed each combination. How many students passed exactly two of the three subjects?

A) 35

B) 45

C) 10

D) 55

Working:

• Exactly two means the three pair-only regions, not the centre.
• 15 (Anatomy and Biochemistry only) + 12 (Anatomy and Physiology only) + 8 (Biochemistry and Physiology only) = 35.

Answer: A) 35

Trap B) 45 adds the centre value of 10, which would be "at least two". Exactly two excludes the triple overlap.

Choosing the correct Venn from statements

The second Venn format gives you statements and asks which diagram fits. Here you reason about how the circles sit, not about numbers. When the circles are nested inside one another or kept completely separate rather than overlapping, they are strictly Euler diagrams, and the UCAT uses these in this format, but the method is identical: translate each statement into a rule about the circles.

Take three statements: all cardiologists are doctors; some doctors are surgeons; no cardiologist is a surgeon. The correct diagram is the only one where the cardiologist circle sits entirely inside the doctor circle, the surgeon circle overlaps the doctor circle, and the surgeon circle does not touch the cardiologist circle. Any diagram that lets cardiologists and surgeons overlap breaks the third statement, and any diagram that puts surgeons entirely inside doctors breaks the word some.

The traps that cost marks

• Double-counting the overlap: |A or B| is |A| + |B| − |A and B|, never just |A| + |B|.
• Only A is not the same as A: the whole circle includes the overlap.
• And versus or: and is the small overlap, or is the large union.
• Forgetting the neither region: the total often exceeds the sum inside the circles.
• Three sets, exactly two versus at least two: exactly two excludes the centre, at least two includes it.
• Choosing a diagram whose overlaps contradict a statement, for example showing an overlap that a "no" statement rules out.
• The word only reverses direction: "only doctors can prescribe" means everyone who prescribes is a doctor, not that every doctor prescribes.

Venn questions often involve a quick total or fraction, so the mental methods in the Quantitative Reasoning complete guide help, and the timings guide keeps you to about a minute each.

Test yourself

The diagram shows 180 patients screened for three symptoms. Answer the three questions, then check the answers below.

1. How many patients had fever in total?
2. How many patients had exactly one symptom?
3. How many patients had at least two symptoms?

Answers

1. 57. The whole fever circle = 30 (only fever) + 12 (fever and cough) + 10 (fever and fatigue) + 5 (all three) = 57.
2. 95. Exactly one symptom = 30 + 25 + 40 = 95.
3. 35. At least two = the three pair-only regions plus the centre = 12 + 10 + 8 + 5 = 35.

Key Takeaway: Fill the overlap first, then the "only" regions, then reconcile against the total. For three sets, work from the centre outward and never let a value sit in two regions. Keep "and" (intersection) and "or" (union) straight, and remember the whole circle always includes the overlap.