Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 630 — #638
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Chapter 15
Cardinality Rules
First, we must determine the sizes of the individual sets, such as P 60 . We can use
a trick: group the 6 and 0 together as a single symbol. Then there is an immediate
bijection between permutations of f0; 1; 2; : : : 9g containing 6 and 0 consecutively
and permutations of:
f60; 1; 2; 3; 4; 5; 7; 8; 9g:
For example, the following two sequences correspond:
.7; 2; 5; 6; 0; 4; 3; 8; 1; 9/ ! .7; 2; 5; 60; 4; 3; 8; 1; 9/:
There are 9Š permutations of the set containing 60, so jP 60 j D 9Š by the Bijection
Rule. Similarly, jP 04 j D jP 42 j D 9Š as well.
Next, we must determine the sizes of the two-way intersections, such as P 42 \
P 60 . Using the grouping trick again, there is a bijection with permutations of the
set:
f42; 60; 1; 3; 5; 7; 8; 9g:
Thus, jP 42 \ P 60 j D 8Š. Similarly, jP 60 \ P 04 j D 8Š by a bijection with the set:
f604; 1; 2; 3; 5; 7; 8; 9g:
And jP 42 \ P 04 j D 8Š as well by a similar argument. Finally, note that jP 60 \
P 04 \ P 42 j D 7Š by a bijection with the set:
f6042; 1; 3; 5; 7; 8; 9g:
Plugging all this into the formula gives:
jP 42 [ P 04 [ P 60 j D 9Š C 9Š C 9Š 8Š 8Š 8Š C 7Š:
15.9.4 Union of n Sets
The size of a union of n sets is given by the following rule.
Rule 15.9.1 (Inclusion-Exclusion).
jS 1 [ S 2 [ [ S n j D
minus
plus
minus
plus
the sum of the sizes of the individual sets
the sizes of all two-way intersections
the sizes of all three-way intersections
the sizes of all four-way intersections
the sizes of all five-way intersections, etc.