Chapter 7. Inclusion-Exclusion aka The Sieve Formula

24 Pages·2017·1.38 MB·English

by  Prof. Tesler

Chapter 7. Inclusion-Exclusion
a.k.a. The Sieve Formula
Prof. Tesler
Math 184A
Winter 2017
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 1 / 24
!"#$%&'($)*+'&
Venn diagram and set sizes
5& 6&
{ }
A = 1, 2, 3, 4, 5
B = {4, 5, 6, 7, 8, 9} /-!-0& ,-.& 1-2-3-4&
{ }
A ∪ B = 1, . . . , 9
{ }
A ∩ B = 4, 5
/7-//-/!-/0-/,-/.&
c { }
(A ∪ B) = 10, . . . , 15
8&
| | | |
A + B counts everything in the union, but elements in the
| |
intersection are counted twice. Subtract A ∩ B to compensate:
| | | | | | | |
A ∪ B = A + B − A ∩ B
9 = 5 + 6 − 2
Size of outside region:
| c| | | | | | | | | | | | |
(A ∪ B) = S − A ∪ B = S − A − B + A ∩ B
6 = 15 − 9 = 15 − 5 − 6 + 2
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 2 / 24
Size of a 3-way union
!"!# !"!#$#!%!# !"!#$#!%!#$#!&!#
"# &# "# %# "# %#
$%# $%# ('# &'# ('# *(# )(# *(#
$%# &'# '(#
$%# ('# ('# )(# )(#
-./0123# *(#
,(#
'# )# &#
(# *# +#
| | | | | | | | | | | |
A A + B A + B + C
2x means the region is counted times.
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 3 / 24
Size of a 3-way union
!"!#!$!#!%!&!"∩!"$!!#'!$!#!%!&!"∩!$"!!&!#"!$∩!#%!!%'!&!"∩$!&!"∩%!&!$∩%!'
"' $' "' $' "' $'
*)' *)' *)' ()' ()' ()' *)' *)' *)'
()' ()' ()'
()' ()' ()' *)' *)' *)'
,-./012' *)' ,-./012' ()' +,-./01' *)'
+)' +)' ()'
%' %' %'
3' 3' 2'
| | | | | | | | | | | | | | | | | |
A + B + C A + B + C A + B + C
| | | | | |
− A ∩ B − A ∩ B − A ∩ B
| | | |
− A ∩ C − A ∩ C
| |
− B ∩ C
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 4 / 24
Size of a 3-way union
∩ ∩ ∩
!"!#!$!#!%!&!" $!&!" %!&!$ %!
∩ ∩
#''!" $ %!'
"' $'
()' ()' ()'
| | | | | | | |
A ∪ B ∪ C = ( A + B + C )
()' | | | | | |
− ( A ∩ B + A ∩ C + B ∩ C )
()' ()'
| |
+ A ∩ B ∩ C
+,-./01' ()'
*)'
%'
2'
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 5 / 24
∩ ∩ ∩
!"!#!$!#!%!&!" $!&!" %!&!$ %!
∩ ∩
#''!" $ %!'
Size of a 3-way union
"' $'
()' ()' ()'
| | | | | | | |
A ∪ B ∪ C = ( A + B + C )
()' | | | | | |
− ( A ∩ B + A ∩ C + B ∩ C )
()' ()'
| |
+ A ∩ B ∩ C
+,-./01' ()' = N − N + N
*)' 1 2 3
%'
2'
where N is the sum of sizes of i-way intersections:
i
| | | | | |
N = A + B + C
1
| | | | | |
N = A ∩ B + A ∩ C + B ∩ C
2
| |
N = A ∩ B ∩ C
3
This is called inclusion-exclusion since we alternately include some
parts, then exclude parts, then include parts, . . .
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 6 / 24
Size of complement of a 3-way union
!"!#$%!&!'!(!'!)!$
∩ ∩ ∩
$$$$$$$$$#!& (!#!& )!#!( )!'!
∩ ∩
& ( )!*$
&$ ($
| c| | | | |
(A ∪ B ∪ C) = S − A ∪ B ∪ C
+,$ +,$ +,$
| |
= S
+,$ | | | | | |
− ( A + B + C )
+,$ +,$
| | | | | |
+ ( A ∩ B + A ∩ C + B ∩ C )
| |
./01234$ +,$ − A ∩ B ∩ C
-,$
)$
"$ = N − N + N − N
0 1 2 3
| |
where N = S .
0
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 7 / 24
Inclusion-Exclusion Formula for size of union of n sets
a.k.a. The Sieve Formula
Inclusion-Exclusion Theorem:
Let A , . . . , A be subsets of a finite set S.
1 n
| |
Let N = S and N be the sum of sizes of all j-way intersections
0 j
(cid:88)
| |
N = A ∩ A ∩ · · · ∩ A for j = 1, . . . , n
j i i i
1 2 j
(cid:54) (cid:54)
1 i <i <···<i n
1 2 j
Then
(cid:88)n
| | j−1
A ∪ · · · ∪ A = N − N + N − N · · · ± N = (−1) N
1 n 1 2 3 4 n j
j=1
(cid:88)n
| c| j
(A ∪ · · · ∪ A ) = N − N + N − N + N · · · ∓ N = (−1) N
1 n 0 1 2 3 4 n j
j=0
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 8 / 24
!"#$%&'$&'('%)'*+"',"*,'
Proof of Inclusion-Exclusion Formula
-' .'
/'
0'
The yellow region is inside k = 2 sets (B and C)
and outside n − k = 3 − 2 = 1 set (A).
Which j-way intersections among A,B,C contain the yellow region?
The ones that only involve B and/or C. None that involve A.
(cid:0) (cid:1)
k
For each j, it’s in j-way intersections:
j
(cid:0) (cid:1)
2
j = 1: = 2 B alone and C alone
1
(cid:0) (cid:1)
2
j = 2: = 1 B ∩ C
2
(cid:0) (cid:1)
2
j = 3: = 0 None
3
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 9 / 24
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Proof of Inclusion-Exclusion Formula
4" 5"
/03" /02" /03"
/01"
/02" /02"
-8%#'9$"
/07"
6"
/03"
:"
A region in exactly k of the sets A , . . . , A is counted in
1 n
N − N + N − · · · this many times (shown for n = 3):
1 2 3
N − N + N
1 2 3
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
k k k
Contribution − + =
1 2 3
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
0 0 0
k = 0 : − + = 0 − 0 + 0 = 0
1 2 3
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
1 1 1
k = 1 : − + = 1 − 0 + 0 = 1
1 2 3
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
2 2 2
k = 2 : − + = 2 − 1 + 0 = 1
1 2 3
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
3 3 3
k = 3 : − + = 3 − 3 + 1 = 1
1 2 3
Prof. Tesler Ch. 7. Inclusion-Exclusion Math 184A / Winter 2017 10 / 24