23.5 Venn diagrams and set notation — Learn

Prerequisites: addition, subtraction and fractions.

Objectives: read and complete Venn diagrams; use set notation for events; calculate probabilities from set counts.

A set is a collection of items that meet a rule. A Venn diagram uses overlapping circles to show sets. The overlap contains items in both sets. The universal set is the full group being considered.

The notation $A\cap B$ means “in A and B”. The notation $A\cup B$ means “in A or B or both”.

Example. In a group of 30 students, 17 study French, 12 study Spanish and 5 study both. Find how many study neither.

Put 5 in the overlap.

French only: $17-5=12$.

Spanish only: $12-5=7$.

In at least one set: $12+5+7=24$.

Neither: $30-24=6$.

So $|F\cup S|=24$ and 6 students are outside both circles.

Watch out. Add the overlap only once when counting “French or Spanish”. Adding 17 and 12 gives 29, which counts the 5 students twice.

Progress check

In a survey of 20 spins, 12 land on red, 9 land on an even-numbered sector, and 5 land on both. Draw a Venn diagram and work out the probability that a spin lands on neither red nor even.

You should now be able to

• represent groups and overlaps in a Venn diagram;
• interpret $A\cap B$ and $A\cup B$;
• calculate probabilities from set counts.