Venn Diagrams — AAI ATC Reasoning Lesson

Before you start

Learning objectives and outcomes

A Venn diagram is just a picture of a relationship between groups. Once you can name the relationship in plain words, the correct diagram almost draws itself.

Objective 1Recognise the five standard diagram shapes used for relationships between three classes.

Objective 2Translate an everyday relationship, such as "some are, none are, all are," into the matching diagram.

Objective 3Read exact counts off a numbered diagram, including for regions where two or three groups overlap.

Objective 4Handle "only," "at least," "at most," and "exactly" correctly when reading a numbered diagram.

Outcome APick the correct diagram for any set of three related classes in under thirty seconds.

Outcome BAnswer a counting question from a numbered diagram without recounting the whole thing from scratch.

Subtopic 1

The five standard diagrams

Almost every three-class relationship in this topic reduces to one of five basic pictures.

Fully separate. No relationship at all — e.g. Table, Elephant, River.

Two disjoint subsets of one. e.g. Animal, Dog, Cat.

Fully nested (a chain). e.g. Kolkata, West Bengal, India.

Two overlapping, one separate. e.g. Teacher, Male, Table.

All three overlapping. e.g. Players, Students, Athletes.

✈ AAI ATC relevance — Venn diagram questions appear in most reasoning sections, usually two to three per shift

Memory trick — "Say the relationship in plain English first"

Before looking at any diagram option, say the relationship between the three classes as a sentence: "all X are Y, but Z is unrelated to either," for example. The right diagram is just that sentence drawn as circles.

EasyATC Pattern

Which standard diagram represents three classes with absolutely no relationship to one another, such as Chair, Mountain and Pen?

With no shared members at all between any pair of classes, three separate non-touching circles is the correct picture.

ModerateATC Pattern

Kolkata, West Bengal and India form a chain where each smaller place lies entirely within the larger one. Which diagram fits?

Since Kolkata is entirely within West Bengal, which is entirely within India, the smallest circle sits fully inside the middle one, which sits fully inside the largest.

Subtopic 2 · highest yield

Choosing the right diagram for three classes

The method is always the same: test each pair of classes for overlap in turn, then combine what you find into one picture.

Test every pair, then combine

For classes A, B and C: ask "do A and B share any members? Do B and C? Do A and C?"
Each yes means those two circles must overlap; each no means they must stay apart.

Three pairwise questions fully determine the diagram, since a Venn diagram is nothing more than a picture of exactly those three relationships at once.

Figure 15.1 — Testing "Teacher, Male, Table" A worked breakdown showing that Teacher and Male overlap (some teachers are male), Male and Table do not overlap (no relationship), and Teacher and Table do not overlap either, leading to the diagram with two overlapping circles for Teacher and Male, and a separate circle for Table. Upload image here

Memory trick — "Three yes-or-no questions decide everything"

You never need to imagine the whole picture at once. Answer the three pairwise overlap questions one at a time, and the complete diagram follows automatically from those three answers.

Worked example

Choose the correct diagram for the classes Players, Students and Athletes.

Players and Students: some players are students, and some students are players, but neither class is entirely inside the other. They overlap partially.
Students and Athletes: likewise, some overlap exists, but neither fully contains the other.
Players and Athletes: many players are athletes and many athletes play, again with partial overlap.
Since all three pairs overlap partially, and it is possible to be a student, a player and an athlete simultaneously, all three circles overlap, with a shared central region for people who belong to all three classes.

ModerateATC Pattern

For the classes Doctor, Woman and Mother, which relationship is correct?

Some doctors are women, some women are mothers, and some doctors are mothers, but none of the three classes is a subset of another and none is unrelated to the others. Three overlapping circles, with a shared central region, is correct.

HardATC Pattern

For the classes Vehicle, Car and Aeroplane, which relationship is correct?

Every car is a vehicle and every aeroplane is a vehicle, so both smaller circles sit fully inside the Vehicle circle. But no car is an aeroplane and no aeroplane is a car, so the Car and Aeroplane circles do not touch each other at all.

Subtopic 3

Reading counts from a numbered diagram

A different style of question gives a completed diagram with a number already written in each region, and asks for counts such as "only A," "exactly two of the three," or "at least one."

Know exactly what each phrase means

"Only A" = the region inside A alone, touching neither B nor C
"A and B only" = the overlap of A and B, excluding the part also inside C
"Exactly two" = add up all three two-way-overlap-only regions
"At least one" = every region inside any circle at all, which is the total minus the outside region

Almost every mistake in this subtopic comes from confusing "A and B" (which includes the all-three region) with "A and B only" (which excludes it).

Figure 15.2 — A numbered three-circle diagram Three overlapping circles labelled Tea, Coffee and Juice, with a number written inside each of the seven regions: only Tea, only Coffee, only Juice, Tea-and-Coffee only, Coffee-and-Juice only, Tea-and-Juice only, and the central region for all three, used for the worked example and practice questions that follow. Upload image here

Memory trick — "Only means alone, not even partly shared"

Whenever a question says "only," picture that region as an island touching no other circle at all. The moment a region is shared with even one more circle, it stops being an "only" region for either class.

Worked example

In a survey of 100 people, a numbered diagram shows: only Tea = 30, only Coffee = 20, only Juice = 15, Tea-and-Coffee only = 10, Coffee-and-Juice only = 8, Tea-and-Juice only = 7, and all three = 5. How many people like exactly two of the three drinks, and how many like at least one?

"Exactly two" means the three two-way-overlap-only regions, excluding the all-three region: 10 + 8 + 7 = 25 people.
"At least one" means every region belonging to any circle at all: 30 + 20 + 15 + 10 + 8 + 7 + 5 = 95 people.
Since the total surveyed was 100, this also tells us 100 − 95 = 5 people liked none of the three drinks, though that number was not directly given.

EasyATC Pattern

In a numbered Venn diagram, the phrase "only A" refers to:

"Only A" is the part of A that belongs to no other class at all — the innermost, unshared slice of that circle.

ModerateATC Pattern

Using the tea, coffee and juice survey above (only Tea = 30, only Coffee = 20, only Juice = 15, Tea-Coffee only = 10, Coffee-Juice only = 8, Tea-Juice only = 7, all three = 5), how many people like Tea in total, including every region that touches the Tea circle?

The total for Tea adds every region touching that circle: only Tea (30) + Tea-Coffee only (10) + Tea-Juice only (7) + all three (5) = 52.

HardATC Pattern

Using the same survey, how many people like exactly one of the three drinks?

"Exactly one" means only the three single-class regions, none of the overlaps: 30 + 20 + 15 = 65.

Exam strategy

Five tips that hold in the AAI ATC hall

Say it in plain English firstTranslate the relationship into a sentence before picking a diagram.

Test every pairThree yes-or-no overlap questions fully determine the diagram.

"Only" means aloneAn "only" region touches no other circle at all.

"A and B" vs "A and B only"The first includes the all-three region; the second excludes it.

Exactly two, exactly oneAdd only the regions matching that exact count, nothing shared beyond it.

At least one = everything insideSum every region that touches any circle at all.

Quick revision

Chapter summary sheet

Five standard shapesSeparate, two subsets of one, fully nested, two overlapping plus one separate, all three overlapping.

MethodTest each pair for overlap, then combine the three answers into one diagram.

Only AThe unshared part of A alone.

A and B onlyShared by A and B, but excludes the all-three region.

Exactly twoSum of all three two-way-overlap-only regions.

At least oneEvery region inside any circle, i.e. total minus "none."

↓ Free download — Venn Diagrams question PDF

Practice choosing the correct diagram for three classes, plus numbered-diagram counting questions with worked numeric answers. Print it, solve it, keep it.

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End of chapter

Comprehensive assessment

Multiple choice questions, building in difficulty. Full solutions are in the downloadable PDF above.

Multiple choice questions

1. Name the five standard diagram shapes used for three-class relationships.

2. Which diagram fits three classes with no relationship at all?

3. Which diagram fits a chain like Kolkata, West Bengal, India?

4. What is the recommended method for choosing the correct diagram?

5. What does "only A" mean in a numbered diagram?

6. What is the difference between "A and B" and "A and B only"?

7. How do you compute "exactly two of three" from a numbered diagram?

8. How do you compute "at least one" from a numbered diagram?

9. In a diagram with 100 people surveyed, if 95 like at least one of three drinks, how many like none?

10. Why does testing each pair of classes fully determine the correct diagram?

Learning objectives and outcomes

The five standard diagrams

Choosing the right diagram for three classes

Test every pair, then combine

Reading counts from a numbered diagram

Know exactly what each phrase means

Five tips that hold in the AAI ATC hall

Chapter summary sheet

↓ Free download — Venn Diagrams question PDF

Comprehensive assessment

Multiple choice questions

Say the relationship in plain English, and the diagram draws itself.