Chapter 12 Venn Diagrams Venn diagrams are illustrations used in the branch of mathematics known as set theory. They show the mathematical or logical relationship between different groups of things (sets). A Venn diagram shows all the possible logical relations between the sets.

Origin John Venn (1834-1923) was a British philosopher and mathematician who introduced the Venn diagram in 1881. A stained glass window in Caius College, Cambridge, where he studied and spent most of his life, commemorates John Venn and represents a Venn diagram.

A Venn diagram of 3 circles.

Real Flowers Fragrant flowers Roses Red flowers

The figure on left is a Venn diagram of 4 ellipses, originally found by Venn himself. See also a black-and-white version, and its Tutte embedding11 (these last two figures provided by Stuart Anderson).

Example 1 Winter Sports In a group of students, 12 like skiing, 10 like snowboarding, 3 like both skiing and snowboarding , 5 don’t like either above How many students are in the group?

Example 2 Music Survey In a poll of 46 students, 23 liked rap music, 24 liked rock music, 19 liked country music, Of all the students, 12 liked rap and country, 13 liked rap and rock, 14 liked country and rock. Of all the students, 9 liked all three types of music. How many did not like any of these types?

Example 3 The Comic Book Collectors Bruce and Clark went to the comic book store to buy some classic comics. They found that they could buy a Batman and a Superman for $18, a Batman and Flash for $16, a Superman and a Flash for $22. How much did each of the comic cost separately?

Example 4 Houses and Apartments Gabby is a real-estate agent who sells a lot of real-estate properties. 2/3 of all the properties that she lists are houses and the remaining are apartments. 1/3 of the houses on the list have swimming pools. 2/3 of those houses with swimming pools also have views. ¼ of all the properties with views have swimming pools. There are 5 apartments with views and there are 2 houses with swimming pools but no view. None of the apartments has a pool.

How many real properties is Gabby selling? How many houses on her list have swimming pool and view? How many apartments on her list do not have a view?