Venn Diagram

What the exam asks. A figure of three or four shapes is given: a triangle, a circle, a rectangle, and sometimes a pentagon, hexagon, oval or square. The question tells you what each shape stands for, for example "the triangle stands for teachers, the circle for writers and the rectangle for painters". Numbers (sometimes letters) are written in the regions, and you are asked something like "How many writers are teachers but NOT painters?"

The logic. A number written in a region counts the persons who are inside exactly those shapes whose outlines contain that region, and in no other shape. So the figure is only a map. Give every region a tag: the letters of the shapes that contain it. Let T be the triangle, C the circle and R the rectangle. The region marked CT is inside the circle and the triangle but outside the rectangle, and the centre CRT is inside all three.

[[fig:0]]

The method: the tag filter. Turn the question into two short lists: the tags that must be present and the tags that must be absent. Keep every region that passes both lists, and add their numbers.

Four shapes. Nothing new is needed; there is only one more tag (S for the square).

[[fig:1]]

In the figure, the region marked CRST is inside all four shapes, the region ST is inside the square and the triangle only, and T is inside the triangle alone. Notice that not every combination is drawn. If no region has the tags R and T without C, then nobody belongs to exactly that group.

The usual slip: "both C and R" also keeps CRT and CRST, not only the region CR. Extra tags are forbidden only when the question says "only", "exactly" or "but not".

Look at the outline, not at the neatness. Shapes in the exam are not always placed tidily. A triangle may stick out above a rectangle, and a circle may cross its edge. A region belongs to a shape only if it lies inside that shape's outline, so the part of a circle that sticks out of the rectangle is "circle, not rectangle".

[[fig:2]]

When letters p, q, r replace the numbers, the same tag method names the region; you only read its letter. Take the numbers as they are given, even if they look odd in real life.

Required tags in, forbidden tags out, then add.

What the exam asks. No figure is given. A short story gives a total, two group sizes and sometimes a "neither" group, and asks for only one, both, neither, or "only A or only B". In the latest papers the story is often just two lines, such as "12 play Football, 8 play Hockey and 5 play both; how many play only Football?" Draw two circles in a rectangle and the picture does the thinking.

Two circles A and B inside a rectangle give four regions: only A, both, only B and neither (the outside of the circles). The four always add up to the whole group N.

The key fact is the union formula:

n(A or B) = n(A) + n(B) − n(A and B)

When you add n(A) and n(B), the people in the overlap are counted twice. Subtracting n(A and B) once puts it right. Total N = n(A or B) + neither.

[[fig:0]]

In the figure, 70 play chess, 65 play carrom and 15 play neither, out of 120. The union is 120 − 15 = 105. Both = 70 + 65 − 105 = 30. Only chess = 70 − 30 = 40 and only carrom = 65 − 30 = 35. Check: 40 + 30 + 35 + 15 = 120.

The order is always the same. First find the union (subtract neither from the total). Second find both. Third find the only-parts. Fourth check that all four regions add up to N.

When percentages are given, treat the whole group as 100 and do the same steps, then convert at the end. When the question links the parts in words ("only bus is 8 more than both"), write the regions with one letter x, and form one equation from the total.

Shortcuts that save a line.

• If everyone is in at least one group (there is no "neither"), both = A + B − N.
• Only A = A − both. Exactly one of the two (only A or only B) = A + B − 2 × both.
• The overlap can never be more than the smaller group, and never less than A + B − N. Use this to cross out options.

Read the wording: "like tea" means all tea lovers, including those who like both. It is not "only tea".

What the exam asks. A story gives three groups (fruits, subjects, hobbies), some set totals, some pair numbers and a centre. The question asks for none, exactly two, only one, all three, or the "any two" group. Some papers give no set totals at all, only some of the buckets (only one, exactly two, all three), and ask for another bucket. Read what is given first, then choose the method.

Three sets bring seven regions, but the working stays calm if you follow one order: from the middle outwards. A, B and C below are the people who like cherries, bananas and apples, out of 100.

The data are: A = 45, B = 40, C = 38; A and B = 15, B and C = 12, A and C = 10; all three = 6. These pair numbers are totals: the 15 includes the 6 who like all three.

[[fig:0]]

1. Centre: all three = 6.
2. Pair-only regions: pair total − centre. A and B only = 15 − 6 = 9, B and C only = 12 − 6 = 6, A and C only = 10 − 6 = 4.
3. Single-only regions: set total − its two pair-only regions − centre. Only A = 45 − 9 − 4 − 6 = 26, only B = 40 − 9 − 6 − 6 = 19, only C = 38 − 6 − 4 − 6 = 22.
4. Union and outside: union = 26 + 19 + 22 + 9 + 6 + 4 + 6 = 92. None = 100 − 92 = 8.

The same union can be found directly with one formula:

n(A or B or C) = n(A) + n(B) + n(C) − n(A and B) − n(B and C) − n(C and A) + n(A and B and C)

The sign pattern is + singles, − pairs, + triple. A person who likes all three is counted 3 times in the singles, taken out 3 times in the pairs, and added back once, so ends up counted once.

The four-bucket check. Every person falls into exactly one of four buckets: exactly one, exactly two, all three and none. So N = (exactly one) + (exactly two) + (all three) + none. When a question gives three of these buckets and asks for the fourth, or gives "only A", "only B" and "only C" together with the rest, you need no formula at all: subtract the known buckets from N.

Read the wording each time. If a number says "like A and B", it includes the centre. If it says "A and B only" or "exactly A and B", the centre is already out. A question on exactly two needs the sum of the three pair-only regions; at least two adds the centre.

What the exam asks. Three words (sometimes four) are given, such as Mother, Woman, Doctor, and the options are four small diagrams. You choose the diagram that shows how the classes are related. In some papers the options are written as statements instead of pictures, for example "All Surgeons are Doctors and some Doctors are Males"; the same pair test decides them. Do not search for the picture. Build it from the pair test, and the correct option will be sitting in front of you.

Take Mother, Woman, Doctor.

• Mother and Woman: every mother is a woman, so Mother is inside Woman.
• Mother and Doctor: a mother can be a doctor, and a doctor need not be a mother, so they overlap.
• Woman and Doctor: a doctor can be a woman, and a doctor can also be a man, so they overlap, and Doctor reaches outside Woman.

Now draw the biggest class first. Woman is a big circle, Mother sits inside it, and Doctor cuts across both Mother and Woman and sticks out on the other side. That is layout 10.

[[fig:0]]

In the exam the four options usually differ in only two facts: who is inside whom and who overlaps whom. So find the one clear "every X is a Y" pair first. It alone throws out most wrong options.

Ask "can one thing be both?": yes means overlap, no means apart, always means inside.

Four words. Four words need four circles, but the method does not change. Run the pair test, and run it carefully, because now there are six pairs. The words usually fall into a tree: one broad class at the top, narrower classes below it, and sometimes a word that does not belong to the tree at all.

Take Food, Cereal, Wheat, Pulse. Wheat is a cereal and a cereal is food, so Wheat is inside Cereal, which is inside Food. A pulse is food, but a pulse is not a cereal. So Pulse is inside Food and apart from Cereal. This is picture 2 below: a big circle holding one nested pair and one separate circle.

[[fig:1]]

Here are the common four-word pictures. 1: four nested circles (Sparrow, Bird, Animal, Living being). 2: the tree we just drew. 3: a big class holding three separate ones (Beverage with Tea, Coffee, Juice). 4: one class holding one member, with two words outside it (Vehicle, Car, Wheel, Engine).

[[fig:2]]

5: two overlapping classes inside a big one and a fourth outside (Person, Teacher, Singer, Table). 6: two separate classes inside a big one and a fourth outside (Bird, Parrot, Crow, Wing). 7: two separate classes with a third cutting both, all inside a big one (Human being, Mother, Father, Doctor). 8: three mutually overlapping classes inside a big one (Person, Teacher, Singer, Writer).

The part-whole idea matters even more here. Wheel, engine, wing, page, petal are parts, not members. A part word is drawn apart from the class it belongs to, unless it is also a member of some larger class in the question.

Find the inside pairs first, hang the other words on the right level, and keep parts outside.

Statements instead of words. Sometimes the words are replaced by sentences, such as "All cats are mammals. No mammal is a bird." The picture method is the same, and the three sentence forms are drawn as shown below. Questions that ask for conclusions from such statements are treated in the Syllogism chapter; here the diagram is chosen.

Three sentence forms are drawn like this.

• All A are B: A inside B.
• Some A are B: A and B overlap.
• No A is B: A and B apart.

Now the key point. Draw only what is stated. Where the sentences are silent, leave the relation open and test the conclusion in every possible picture. A conclusion follows only if it is true in every picture that the statements allow.

Take: All dogs are mammals. All mammals are warm-blooded. All birds are warm-blooded.

[[fig:3]]

Dogs sit inside mammals, which sit inside warm-blooded animals. Birds are also inside warm-blooded animals, but nothing says they are mammals, so they are drawn apart from mammals. Now read: "All dogs are warm-blooded" is true in the picture, so it follows. "All warm-blooded animals are mammals" is false, because birds are warm-blooded and not mammals, so it does not follow.

When the statements leave a relation open, there are many pictures.

[[fig:4]]

From "All teachers (T) are graduates (G)" and "Some graduates are artists (A)", the artists may or may not touch the teachers. So "Some teachers are artists" is only a possibility, not a conclusion. A conclusion follows only if it holds in every possible picture.

What the exam asks. The question turns around. A diagram is shown, without names, and the four options are groups of three words. The paper says "Select the set of classes the relationship among which is best illustrated by the following Venn diagram". The words are the same kind of words as in Type 4; only the direction is reversed.

Start by reading the picture as a set of statements. Name the circles X, Y, Z. In the figure below:

• Y is inside X (every Y is an X).
• Z overlaps X, but not Y (some Z are X, no Z is Y).
• Y and Z are apart.

[[fig:0]]

Now test each option, word by word, with the pair test. An option is correct only if it passes every pair. One failed pair is enough to throw it out.

Take Cow, Animal, Carnivore. A cow is an animal, so Cow is inside Animal. A carnivore can be an animal (a tiger), so they overlap. No cow is a carnivore, so Cow and Carnivore are apart. All three facts match the picture, so this option fits.

The usual failure is to look at sizes instead of overlaps. A circle drawn small does not mean the class is small. Only the positions matter: inside, cutting, apart. Read the picture as statements, then test each option pair by pair.

The quickest way is to match the option with the layout number, then eliminate. If you have learnt the ten layouts, the right option often appears at once.

The favourite words of the paper-setters. In recent papers the same word families keep coming back: relation words (Mothers, Sisters, Daughters; Fathers, Males, Sons), a quality with two genders (Girls, Intelligent, Boys), and a class with its kinds (Wheat, Maize, Cereals; Peso, Currency, Taka). Three pictures cover most of them. Three circles all overlapping is the picture of Mothers, Sisters, Daughters, because one woman can be all three. One circle cutting two separate circles is the picture of Girls, Intelligent, Boys. Two separate circles inside a bigger one is the picture of Wheat, Maize, Cereals.