Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 631 — #639
15.9. Inclusion-Exclusion 631
The formulas for unions of two and three sets are special cases of this general
rule.
This way of expressing Inclusion-Exclusion is easy to understand and nearly
as precise as expressing it in mathematical symbols, but we’ll need the symbolic
version below, so let’s work on deciphering it now.
We already have a concise notation for the sum of sizes of the individual sets,
namely,
nX
jS i j:
iD1
A “two-way intersection” is a set of the form S i \ S j for i ¤ j . We regard S j \ S i
as the same two-way intersection as S i \ S j , so we can assume that i < j . Now
we can express the sum of the sizes of the two-way intersections as
X
jS i \ S j j:
1i