Reasoning topic

Venn diagram reasoning: which circles fit the words

These questions give a few words and ask which diagram of circles shows how their groups sit together. Others give the size of each group and ask you to count.

What the question looks like

A typical question asks which diagram shows how squares, rectangles and circles relate as groups.

This question gives three items. Using the relationship between them, choose the most suitable diagram from those given below. Squares, Rectangles, Circles.

  1. Three circles, one inside the next
  2. Two separate circles inside a larger circle
  3. Three separate circles
  4. One circle inside another, and a third circle apart
Answer

One circle inside another, and a third circle apart

Every square is a rectangle, so the squares circle goes inside the rectangles circle. No circle is a square or a rectangle, so its circle stays apart.

How it works

Take the words two at a time. Ask: is every A a B? Is every B an A? Can something be both?

Every A is a B: A goes inside B. Some things are both, but neither group holds the other: the circles overlap. Nothing is both: the circles stay apart.

To count, add the groups and take the overlap out once: at least one = A + B − both.

This question gives four items. Choose the diagram that best represents the relationship between them. Living things, Animals, Dogs, Stones.

  1. Every dog is an animal, but not every animal is a dog. Dogs go inside animals.
  2. Every animal is a living thing, but not the other way round. Animals go inside living things.
  3. Stones: a stone is not a living thing, so its circle stays apart from all three.
  4. The picture is three circles, one inside the next, with a fourth circle apart.

Answer: Three circles one inside the next, and a fourth circle apart

Try three

1. This question gives five items. Choose the diagram that best represents the relationship between them. Vehicles, Cars, Birds, Parrots, Rivers.

  1. Three circles one inside the next, and a pair one inside the other
  2. Two pairs, each one circle inside another, and a fifth circle apart
  3. Two separate circles inside a larger circle, and two more circles apart
  4. Five circles in a row, each overlapping the next
Hint

Which words pair up, and which one stands alone?

Answer

Two pairs, each one circle inside another, and a fifth circle apart

Every car is a vehicle and every parrot is a bird, so each pair is one circle inside another. No vehicle is a bird, so the two pairs stay apart. A river is none of the other four, so its circle stands alone.

2. This question gives three items. Using the relationship between them, choose the most suitable diagram from those given below. Days, Rainy days, Saturday

  1. Three separate circles
  2. Two overlapping circles inside a larger circle
  3. Two separate circles inside a larger circle
  4. Three circles in a chain, the middle one overlapping both ends
Hint

Test each pair both ways round.

Answer

Two overlapping circles inside a larger circle

Every rainy day is a day, and every Saturday is a day, so both circles go inside the circle for days. A Saturday can be rainy, so those two circles overlap inside it.

3. In a survey of 200 students, 120 like cricket, 95 like football and 45 like both. How many students like neither cricket nor football?

  1. 30
  2. 75
  3. 170
  4. 45
Hint

The 45 who like both are inside both totals.

Answer

30

120 + 95 = 215 counts the 45 who like both twice. Take them out once: 215 − 45 = 170 like at least one game. So 200 − 170 = 30 like neither.

Venn Diagrams in the app

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The concepts behind it

In the official syllabus of

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