Reading Venn Diagrams

Grade 4

What a Venn diagram shows

A Venn diagram uses overlapping circles inside a rectangle to organise things into groups. Each circle is a set — a collection of items that share a property.

The overlap (the lens-shaped middle) contains items that belong to both sets. Items outside all circles but still inside the rectangle belong to neither set. The rectangle itself represents everything being considered — this is called the universal set, labelled ξ (or sometimes U or ε).

Filling in a Venn diagram

EXAMPLE

In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. How many study neither? Fill in the Venn diagram and find P(randomly chosen student studies French only).

Start with the overlap: 5 study both — write 5 in the middle section. French only: 18 − 5 = 13. Spanish only: 12 − 5 = 7. Total accounted for: 13 + 5 + 7 = 25. Neither: 30 − 25 = 5. P(French only) = 13/30.

Every number belongs in exactly one region

When filling in a Venn diagram, always start with the overlap and work outwards. If you place the full group total in the circle first and then try to subtract, you'll count overlapping students twice.

Reading probabilities off the diagram

Once the diagram is filled in, every probability question becomes a reading exercise. To find the probability of a randomly chosen person being in a region, count the number in that region and divide by the total in the universal set.

The numbers in all regions of the diagram must add up to the total in the universal set. Use this as a check — if they don't add up, you've misplaced a value.

Try this one

For two sets \( A \) and \( B \), \( n(A) = 20 \), \( n(B) = 14 \) and \( n(A \cup B) = 27 \).

(a) Work out \( n(A \cap B) \). [2]

(b) Work out \( n(A \cap B') \), the number in \( A \) only. [2]

Nova's hint:

This uses the addition rule connecting the sizes of two sets, their union and their overlap.

Ask yourself: 'when I add \( n(A) \) and \( n(B) \), how many times is the overlap counted?' Rearranging the rule gives the size of the intersection.

Try to Solve it with Nova?