Venn diagrams

How many children like maths, how many like physics, and how many like both? Such questions get confusing as soon as groups overlap. A Venn diagram brings order by drawing each set as a circle and making the overlap visible.

Two circles, four regions

Draw two overlapping circles for the sets $A$ and $B$, both inside a rectangle that holds every object you are looking at. Inside the circles this creates three regions: only $A$, only $B$, and the overlap in the middle standing for $A \cap B$. On top of that comes the region outside the circles for everything that belongs to neither set. Each object lands in exactly one of these four regions.

Filling the regions with counts

When counting, always start in the middle. First enter how many objects lie in both sets. Then subtract this number from the totals of the individual sets so you do not count them twice. That way every region gets its own count.

Turning words into a diagram

If a task says 14 children play football, 8 play tennis and 5 play both, then the 5 goes in the middle. The silent convention behind this matters: those 5 children are already included in the 14 and in the 8, because they do play football and they do play tennis. So only football is played by $14 - 5 = 9$ children, only tennis by $8 - 5 = 3$ children. That turns running text into a clear picture with three numbers.

What lies outside

Often there are objects belonging to no set. You find their count by taking the total number of objects and subtracting all three regions of the circles. What remains stands outside the circles in the surrounding rectangle.