Combinatorics Through Guided Discovery, 2004a

Chapter 5
The Principle of Inclusion and
Exclusion
5.1 The size of a union of sets
One of our very first counting principles was the sum principle which says that
the size of a union of disjoint sets is the sum of their sizes. Computing the size
of overlapping sets requires, quite naturally, information about how they overlap.
Taking such information into account will allow us to develop a powerful extension
of the sum principle known as the “principle of inclusion and exclusion.”
5.1.1 Unions of two or three sets
◦ Problem 225. In a biology lab study of the effects of basic fertilizer ingredients
on plants, 16 plants are treated with potash, 16 plants are treated with
phosphate, and among these plants, eight are treated with both phosphate
and potash. No other treatments are used. How many plants receive at least
one treatment? If 32 plants are studied, how many receive no treatment?
+ Problem 226. Give a formula for the size of the union A ∪ B of two sets A
in terms of the sizes |A| of A, |B| of B, and |A ∩ B| of A ∩ B. If A and B
are subsets of some “universal” set U, express the size of the complement
U − (A ∪ B) in terms of the sizes |U| of U, |A| of A, |B| of B, and |A ∩ B| of
A ∩ B. (h)
◦ Problem 227. In Problem 225, there were just two fertilizers used to treat
the sample plants. Now suppose there are three fertilizer treatments, and
15 plants are treated with nitrates, 16 with potash, 16 with phosphate, 7
93