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Logical Venn Diagrams

Reasoning · Logical & Analytical Reasoning

7 Part1 worked examples11 practice questions

Suppose three words are put in front of you — doctors, teachers, humans. Each word names a group (a class) of people. The question is how the three groups are related. Is every doctor a human? Can a doctor also be a teacher? The art of showing this relation with circles is called a Venn diagram.

The method was made popular by the English logician John Venn (1834–1923), so it carries his name. Each group gets one circle. The size of the circle does not matter — what matters is its position: inside another circle, cutting it, or away from it.

This chapter gives four kinds of questions in exams. We will learn them one by one from scratch:

Type of questionWhat is askedWhere we learn it
Three words → diagram“Which picture shows these three classes?”Parts 1 and 2
Picture → region“Which number shows those who are …?”Part 3
Numbers in figures“Which number lies in both the circle and the triangle?”Part 3
Counting“How many people drink only tea?”Part 4

How to use this chapter

  1. Read and look at the picture — every idea comes with a figure. Look at it closely; the answer lives in the picture.
  2. Study the solved example — the full solution is open. At each step ask: why this step?
  3. Try it yourself — in the “Try it yourself first” questions the solution is hidden. Think on paper, choose an option; pick again if wrong. If stuck take the hint, and open the solution last.
  4. Watch the level — Basic, Standard, Advanced. Make the basic ones solid, then move on.
NoteLetters (A, B, C …) and numbers in the figures are the same in both languages. Which circle stands for which class is written in the words of the question.
1

Foundation — circle, class and the relation of two classes

Only three possibilities: inside, cutting, apart

What a class and a circle mean, the three possible relations between two classes and the test to tell them apart.

A class and its circle

One group = one circle

What a class (group) is, how a circle shows it and the three possible pictures of two classes.

A class is a group of things or people that share some property: fruits, students, rivers. Each class gets one circle. Everything inside the circle is a member of that class and everything outside is not.

Between two classes there are only three possible relations (when they do not mean the same group):

YX
Inside — every member of X is also in Y (e.g. X = cows, Y = animals)
XY
Cutting — some members are in both, some in only one (e.g. X = doctors, Y = teachers)
XY
Apart — no member is in both (e.g. X = dogs, Y = cats)

There is a fourth case: if two words name the same group (like “rupee” and “the currency of India”), they share one circle. It is rare in exams.

NoteDo not look at the size of the circle. The circle of “teachers” is drawn smaller than “humans” because it sits inside it, not because there are fewer teachers. You only decide: inside, cutting or apart.
Example 1 · Basic
Which picture suits the two classes “cows” and “animals”?
Solution, step by step
  1. 1Ask: is every cow an animal? Yes.
  2. 2So the whole group of cows is a part of the group of animals.
  3. 3So the circle of cows goes inside the circle of animals.
Answer The first picture (inside)
“Every X is a Y” means X’s circle inside Y’s.
BasicTry it yourself first
Two circles are drawn for Roses and Flowers. Which description of the drawing is right?
BasicTry it yourself first
Two circles are drawn for Aquatic animals and Mammals. Which description of the drawing is right?
BasicTry it yourself first
Two circles are drawn for Lions and Wild animals. Which description of the drawing is right?
BasicTry it yourself first
Two circles are drawn for Foreigners and Doctors. Which description of the drawing is right?
BasicTry it yourself first
Two circles are drawn for Rice and Cereals. Which description of the drawing is right?

The test that tells the relation

Three short questions, always in the same order

Three questions to fix the relation of two classes, the role of the word “possible” and the common slip.

To decide the picture of two classes X and Y ask these three questions in order:

Three questions that fix the relation
1. Is every member of X in Y? Yes → X’s circle inside Y’s2. (No) Is every member of Y in X? Yes → Y’s circle inside X’s3. (No) Can even one thing possibly be in both? Yes → cutting; No → apart

In the third question the word “possibly” is the key. We do not check whether something is in both today; we check whether it is possible. One person can be a doctor and a teacher, so the circles cut. But no animal can be a dog and a cat at once, so the circles stay apart.

XYTestPicture
TigersWild animalsevery tiger is a wild animalinside
SingersWomenone person can be bothcutting
CarsRiversno car is a riverapart
ChildrenFathersa child is not a fatherapart
NoteThere are two opposite slips. (1) Drawing separate circles when “some X are Y” — sharing some members is exactly what “cutting” means. (2) Drawing cutting circles when every X is a Y — then X lies wholly inside Y.

Now test some pairs yourself. Each time ask the three questions above in order.

StandardTry it yourself first
Two circles are drawn for Boys and Students. Which description of the drawing is right?
StandardTry it yourself first
Two circles are drawn for Teachers and Poets. Which description of the drawing is right?
StandardTry it yourself first
Two circles are drawn for Iron and Liquids. Which description of the drawing is right?
StandardTry it yourself first
Two circles are drawn for Women and Rivers. Which description of the drawing is right?
AdvancedTry it yourself first
Two circles are drawn for Indians who know English and Indians. Which description of the drawing is right?
AdvancedTry it yourself first
Two circles are drawn for Women and Teachers. Which description of the drawing is right?
2

Three classes — choosing the right diagram

Relation pair by pair, then the picture

The three pairs of three words, eleven standard pictures and the confusing cases.

Three words: the pair-by-pair method

Three words = three pairs

The steps to fix the picture of three classes and the first solved example.

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Plain pictures: nested, separate and one cutting pair

Six standard pictures

Three nested circles, three separate ones, two circles in a big one, and two more combinations.

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Mixed pictures: when the third circle cuts

Five harder pictures

All three cutting, the grandfather–male–child type, the sister–woman–teacher type, the boys–girls–students type and the “combination of two conditions” type.

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The confusing cases

Where most mistakes in choosing the picture happen

Tricky word groups, the “combination of two conditions”, the “possible” test and the common slips.

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3

Reading a diagram — regions and figures

Now the picture is given, not the words

Naming the seven regions of three circles and reading numbers placed in a circle, square and triangle.

The seven regions of three circles

Every region is a different kind of group

What each region of three cutting circles means, how to read “and / or / not / only”, and counting compartments.

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Numbers inside a circle, a square and a triangle

For each number see which figures it is inside

The method for reading numbers placed in overlapping figures of different shapes.

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4

Counting — getting numbers out of a Venn diagram

Every person in exactly one region

Counting questions with two and three groups, percents, and reading numbers printed in a figure.

Counting with two groups

Do not count the “both” group twice

The picture, the formula and four common slips for questions with two groups.

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Counting with three groups

Seven regions, fill from the centre outwards

Counting with three groups: the order of filling, the formula and “exactly one / exactly two / at least two”.

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Percents and numbers printed in the figure

Imagine a group of 100; choose regions, then add

Percent counting questions and questions on figures where each region prints its headcount.

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5

Strategy guide

Spot the type of question at once

The first step for each type, time-savers and common mistakes.

Spot the type, then pick the method

Where to begin

A table for picking the right method from the look of the question.

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Common mistakes and time-savers

The mistakes that keep happening

The commonest reasons for losing marks in Venn questions and how to avoid them.

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6

Mixed practice

All types mixed — first spot the type, then solve

Questions of different types together, in three levels.

Basic practice

One idea, one step

Pair relations, plain diagrams, regions and simple counting.

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Standard practice

Confusing pictures, thinking about the centre, three-group counting

Mixed diagrams, “and / not” regions, counts of exactly one and so on.

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Advanced practice

Fine-difference pictures, hard conditions, long counts

The finest diagrams, regions like “exactly two / at least two”, and long three-group counts.

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7

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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