POP

G5.23 Venn Diagrams

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01 — Explanation

What do the set symbols actually mean?

Symbol	Meaning	Example ( $\xi=\{1..5\}$ , $A=\{1,2\}$ , $B=\{2,3\}$ )
$\xi$	universal set (everything)	$\xi=\{1,2,3,4,5\}$
$n(A)$	number of elements in $A$	$n(A)=2$
$\cap$	intersection — AND	$A\cap B=\{2\}$
$\cup$	union — OR	$A\cup B=\{1,2,3\}$
$A'$	complement — NOT in $A$	$A'=\{3,4,5\}$
$\varnothing$	empty set	—

02 — Explanation

How do I shade a complex region like (A\cup B)'?

Treat it like order of operations: deal with the brackets first, then apply the outer operator.

$(A\cup B)$ = everything inside either circle.
Apply $'$ (NOT) to that whole region = everything outside both circles.

Similarly, $A\cap B'$ means: take circle $A$ , take everything outside $B$ , then find where those two regions overlap — i.e. the part of $A$ that does not overlap with $B$ .

Exam Tip

Never try to shade a compound expression in one step — breaking it into the bracket first, then the outer operation, removes almost all ambiguity.

03 — Explanation

How do I fill in a Venn diagram from a word problem?

Work from the inside out — start with the most specific (most constrained) piece of information and subtract outward.

Place the value for the all-sets intersection first (the centre).
For each two-set overlap, subtract the centre value to get that pair's "only" region.
For each single set, subtract all its overlap regions to get its "only" region.
Find the "outside everything" region by subtracting the sum of all placed values from $n(\xi)$ .
Check: every region should sum to exactly $n(\xi)$ .

Exam Tip

A very common error is placing a number like "17 have sheep and chickens" directly into the "sheep and chickens only" region — but it actually represents the entire overlap, including anyone who also has a third thing. Always subtract any central/triple overlap from it first.

04 — Explanation

How do I calculate a basic probability from a completed diagram?

P(A) = \frac{n(A)}{n(\xi)}

For unions or intersections, use the number of elements in that specific region as the numerator, over the same universal total.

05 — Explanation

How do I handle "given that..." conditional probability?

The phrase "given that" restricts your sample space — the denominator changes from the universal total to the size of the given condition.

Method: new denominator = size of the given set; numerator = the overlap between the given set and the set you want.

Worked Example

In a survey of 50 people, 30 own a cat (C), 25 own a dog (D), and 6 own neither.

Union: $50-6=44$ own at least one.
Intersection: $n(C)+n(D)-n(C\cup D) = 30+25-44=11$ .
"Cat only": $30-11=19$ . "Dog only": $25-11=14$ .
Check: $19+11+14+6=50$ . ✓
$P(C\cap D)=\dfrac{11}{50}$ .
Given that a person owns a cat, find $P(\text{also owns a dog})$ : new denominator is $n(C)=30$ ; numerator is the overlap, $11$ .

P(D\mid C)=\frac{11}{30}

Exam Tip

Using the universal total (50) instead of the restricted one (30) in a conditional question is the single most common error on this topic — actively look for "given that" and ask "what is my new total?" before calculating.