A rectangle stands for the universal set, and a circle inside it stands for each set. Where two circles overlap, that patch is the intersection. Nothing new is being introduced: the four regions you counted last topic are simply being drawn.
On a Venn diagram the four groups become four patches of paper, and a patch cannot be forgotten because you can see it sitting there empty. The mathematics is identical; the diagram just stops you making bookkeeping mistakes.
The parts of the diagram
| Part | What it stands for |
|---|---|
| The rectangle | The universal set U |
| A circle | One set, labelled with its letter |
| The overlap | A ∩ B: the things in both sets |
| Circle A outside the overlap | A − B: in A only |
| Inside the rectangle, outside both circles | Neither set. The region everybody forgets |
The space inside the rectangle but outside every circle is a real region with real members, and it is the one students leave blank. If your four regions do not add up to n(U), that corner is almost always where the missing members went.
Filling in a Venn diagram
There is one rule that makes these questions easy: always start in the middle. Put the number who are in both sets into the overlap first, then work outwards.
When a question says 25 study science, that 25 includes the ones who also study computing. It is a total for the whole circle, not for the crescent outside the overlap.
Reading a shaded diagram
| What is shaded | What it means |
|---|---|
| Both circles completely | A ∪ B |
| Only the overlap | A ∩ B |
| Circle A with the overlap left white | A − B |
| Everything except circle A | A′ |
| The rectangle corners only | (A ∪ B)′, meaning neither |
A hospital comparing patients with two symptoms, a school working out who takes both optional subjects, a survey of who has electricity and who has piped water. The diagram makes the overlap impossible to overlook, and the overlap is exactly what people get wrong.
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In a group of 50, 30 read the newspaper, 22 listen to the radio, and 12 do both. Find how many do neither.
In a class of 35, 20 play chess, 5 play neither, and 4 play both. How many play carrom?
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A Venn diagram draws sets as circles inside a rectangle standing for the universal set.
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