Blank Venn diagram

Pick two, three or four sets. Leave a label empty and the sheet prints a line there for a pupil to write on, so the default is a blank template ready for any topic.

Open Write items in the regions to fill one in. Each box is named after the labels you typed, such as Cats only or Cats and Dogs only, and the words are centred in the widest part of that region. Long lists get smaller type. Tick A box for the universal set to draw the rectangle round the outside, with a box for things that belong to none of the sets.

Print gives the page as shown, a blank copy with the labels kept, or both, one after the other. That's handy for a worked example on one side of the desk and a fresh one to try. Landscape suits most diagrams. Download SVG saves the same drawing for slides or a document, sized in millimetres for the paper you picked.

A Venn diagram must show every combination of sets, even the empty ones. With n sets that's 2ⁿ regions counting the outside: 4 for two sets, 8 for three, 16 for four.

Circles run out at four. Two circles cross at two points at most, and n circles can cut the page into no more than n² − n + 2 pieces. For four circles that's 14, two short of 16, so some pair of sets would have no region of its own however you arrange them.

Ellipses get round it because they can be long and thin and cross each other four times. John Venn drew a four-set version with ellipses himself. The one here leans two outer ellipses in towards two that cross in the middle, and the tests on this page check all 15 inside regions are there.

A diagram that only draws the regions that can happen is an Euler diagram instead. Show cats and dogs as two separate circles and you've drawn an Euler diagram, because nothing is both.

John Venn (1834–1923) introduced the diagrams in July 1880, in a paper called "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings" in the Philosophical Magazine. He called them Eulerian circles, after the Swiss mathematician Leonhard Euler. The name Venn diagram came later; Wikipedia credits C. I. Lewis with it in A Survey of Symbolic Logic, 1918.

Gonville and Caius College, Cambridge, where Venn was a fellow and later president, has a stained-glass window in its dining hall that commemorates him with a Venn diagram.