Venn Diagrams

A Venn diagram sorts items into overlapping circles, making it easy to see which belong to one group, both groups, or neither. They are one of the clearest ways to organise probability information, and OCR uses them at both Foundation and Higher tier. This lesson covers the set notation ( $\cup$ , $\cap$ , $'$ , $\xi$ ), how to fill in and read two-set and three-set Venn diagrams, and how to find probabilities from them.

This is AO1 (using set notation and completing diagrams), AO2 (interpreting regions) and AO3 (multi-set reasoning and conditional probability from a Venn). OCR command words include Complete, Work out, Write down and Find.

Key Vocabulary

Symbol / Term	Meaning
$\xi$	The universal set — every item being considered.
$A \cup B$	Union — in $A$ or $B$ or both.
$A \cap B$	Intersection — in both $A$ and $B$ .
$A'$	Complement — not in $A$ .
$n(A)$	The number of items in set $A$ .
Region	A part of the diagram, e.g. " $A$ only" or "neither".

Set Notation

Before drawing diagrams, learn the four key symbols:

$\cup$ (union) means "or" — everything in either circle, including the overlap.
$\cap$ (intersection) means "and" — only the overlap.
$A'$ (complement) means "not $A$ " — everything outside circle $A$ .
$\xi$ (the universal set) is the rectangle containing everything.

For example, $(A \cup B)'$ means "not in $A$ or $B$ " — the region outside both circles, which is the same as $A' \cap B'$ . Being able to translate a piece of notation into the region it describes is half the battle: $A \cap B'$ is "in $A$ but not $B$ " (the $A$ -only region); $A' \cap B$ is "in $B$ but not $A$ " (the $B$ -only region); and $A' \cap B'$ is "in neither". Practise saying each symbol in words and pointing to the matching part of the diagram, and the harder combinations stop being intimidating.

Notice that the combinations come in natural pairs through the complement. The opposite of "in $A$ or $B$ " is "in neither", and the opposite of "in both" is "not in both". This is why the complement region — outside the circles — appears so often: many questions that look complicated are really asking for "everything except" some region, which you find by subtracting from the universal set $\xi$ .

Worked Example 1

For two sets $A$ and $B$ , describe in words: (a) $A \cap B$ , (b) $A \cup B$ , (c) $A'$ .

Solution: (a) $A \cap B$ = items in both $A$ and $B$ (the overlap). (b) $A \cup B$ = items in $A$ or $B$ or both. (c) $A'$ = items not in $A$ .

Common error: Swapping $\cup$ and $\cap$ . A handy memory aid: $\cup$ looks like a cup that holds everything; $\cap$ is the smaller cap, just the overlap.

Two-Set Venn Diagrams

A two-set Venn diagram has four regions: $A$ only, the intersection $A \cap B$ , $B$ only, and neither $(A \cup B)'$ . Always fill in the intersection first, then work outwards.

The reason for filling the intersection first is that the totals you are usually given — " $18$ study French", " $12$ study Spanish" — include the overlap. If you wrote $18$ in the French circle and $12$ in the Spanish circle, the students who study both would be counted in both numbers, and your regions would add up to more than the class. By placing the "both" figure in the overlap first and then subtracting it from each total, you get the "only" regions correctly, and everything adds up. This subtract-the-overlap step is the heart of the technique and the place most marks are won or lost.

Once the four regions are filled, every question becomes a matter of reading the right regions and dividing by the right total. "French only" is a single region; " $F \cup S$ " (French or Spanish) is the three regions inside the circles; " $F \cap S$ " (French and Spanish) is just the overlap; and "neither" is the region outside both circles. Always check your four regions add up to the universal set $\xi$ before computing any probability — it is a free error-check that catches most slips.

Worked Example 2

In a class of $30$ students, $18$ study French ( $F$ ), $12$ study Spanish ( $S$ ) and $5$ study both. Complete a Venn diagram and find the number who study neither.

Solution: Start with the intersection $n(F \cap S) = 5$ .

French only $= 18 - 5 = 13$ .
Spanish only $= 12 - 5 = 7$ .
Neither $= 30 - (13 + 5 + 7) = 5$ .

The completed diagram is shown below.

Check: $13 + 5 + 7 + 5 = 30$ . ✓ So $5$ students study neither language.

Worked Example 3

Using the diagram in Worked Example 2, a student is chosen at random. Work out: (a) $P(\text{studies French only})$ (b) $P(F \cup S)$ (c) $P(F \cap S)$

Solution: (a) French only $= 13$ , so $P = \dfrac{13}{30}$ . (b) $F \cup S = 13 + 5 + 7 = 25$ , so $P(F \cup S) = \dfrac{25}{30} = \dfrac{5}{6}$ . (c) $F \cap S = 5$ , so $P(F \cap S) = \dfrac{5}{30} = \dfrac{1}{6}$ .

Worked Example 4

In a survey of $40$ shoppers, $22$ bought bread ( $B$ ), $18$ bought milk ( $M$ ) and $10$ bought both. Work out $P(B \cup M)$ .

Solution: Bread only $= 22 - 10 = 12$ ; milk only $= 18 - 10 = 8$ ; both $= 10$ .

$P(B \cup M) = \dfrac{12 + 10 + 8}{40} = \dfrac{30}{40} = \dfrac{3}{4}$

This agrees with the addition rule for overlapping sets: $P(B) + P(M) - P(B \cap M) = \dfrac{22}{40} + \dfrac{18}{40} - \dfrac{10}{40} = \dfrac{30}{40}$ .

Common error: Adding $22 + 18 = 40$ and forgetting to subtract the overlap; the $10$ "both" shoppers would be counted twice.

Reading the Complement and "Only" Regions

The "only" regions and the complement trip up many students, so it pays to be deliberate. " $A$ only" means inside circle $A$ but outside the overlap, whereas " $A$ " on its own includes the overlap. Likewise $A'$ (not $A$ ) is everything outside circle $A$ — both the " $B$ only" region and the "neither" region. Reading the wording carefully, and pointing to the exact region on the diagram, prevents the common slip of including or excluding the overlap by mistake.

Worked Example 5

Using the shopper diagram (Worked Example 4, total $40$ ), work out: (a) $P(B')$ — does not buy bread. (b) $P(\text{bought neither})$ .

Solution: Neither $= 40 - 30 = 10$ . (a) $B' =$ everything outside the bread circle $=$ milk only $+$ neither $= 8 + 10 = 18$ , so $P(B') = \dfrac{18}{40} = \dfrac{9}{20}$ . (Check: $1 - \dfrac{22}{40} = \dfrac{18}{40}$ . ✓) (b) $P(\text{neither}) = \dfrac{10}{40} = \dfrac{1}{4}$ .

Three-Set Venn Diagrams

A three-set Venn diagram has eight regions. The safest order is to fill in the central region ( $A \cap B \cap C$ ) first, then the three "two-set only" overlaps, then the "one-set only" regions, and finally "neither".

The reason for working from the centre outwards is the same as for two sets, but the bookkeeping is harder, so a disciplined order is essential. A figure such as " $12$ like tea and coffee" almost always includes the people who like all three, so you must subtract the centre to get the "tea and coffee only" region. Likewise, a total such as " $25$ like tea" includes everyone in the tea circle — the tea-only region plus the two pairwise overlaps that touch tea plus the centre — so you subtract all three of those to find tea-only. Skip the centre-first step and the overlaps will be wrong, which throws off every region after them.

Venn Diagrams

Venn Diagrams

Key Vocabulary

Set Notation

Worked Example 1

Two-Set Venn Diagrams

Worked Example 2

Worked Example 3

Worked Example 4

Reading the Complement and "Only" Regions

Worked Example 5

Three-Set Venn Diagrams

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