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Unformatted text preview: 64 1. ALGEBRAIC THEMES (a)
jA0 \ A0 \
1
2 \ A0 j D jU j
n X jAi j C i X X jAi \ Aj j i <j C . 1/n jA1 \ An j: C . 1/n jAi \ Aj \ Ak j C An j: i <j <k (b)
jA1 [ A2 [ [ An j D X jAi j i C X X jAi \ Aj j i <j jAi \ Aj \ Ak j 1 jA1 \ i <j <k P
Proof. For any subset X of U , jX j D u2U 1X .u/. The desired equalities are obtained by starting with the identities for characteristic functions
given in the proposition, evaluating both sides at u 2 U , and summing
over u.
I
The formulas given in the corollary are called the inclusionexclusion
formulas.
Example 1.9.13. Find a formula for the number of permutations of n
with no ﬁxed points. That is, is required to satisfy .j / ¤ j for all
1 Ä j Ä n. Such permutations are sometimes called derangements. Take
U to be the set of all permutations of f1; 2; : : : ; ng, and let Ai be the set of
permutations of f1; 2; : : : ; ng such that .i / D i . Thus each Ai is the set
of permutations of n 1 objects, and so has size .n 1/Š. In general, the
intersection of any k of the Ai is the set of permutations of n k objects,
and so has cardinality .n k /Š. The situation is ideal for application of the
inclusionexclusion formula because the size of the intersection of k of the
Ai does not depend on the choice of the k subsets. The set of derangements
is A0 \ A2 \ \ A0 , and its cardinality is
n
1
Dn D jA0 \ A0 \
1
2 \ A0 j D jU j
n X jAi j C i X jAi \ Aj \ Ak j C X jAi \ Aj j i <j C . 1/n jA1 \ An j: i <j <k ÂÃ
n
As each k fold intersection has cardinality .n k /Š and there are
such
k
intersections, Dn evaluates to
ÂÃ
ÂÃ
n
kn
Dn D nŠ n.n 1/ŠC
.n 2/Š : : : C. 1/
.n k /ŠC C. 1/n :
2
k ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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