Sum Rule
Mathematics for Computer Science
MIT 6.042J/18.062J
|A B| = |A| + |B|
Inclusion-exclusion
Counting practice
A
B
for disjoint sets A, B
Albert R Meyer,
April 16, 2010
lec 10F.1
Sum Rule
Albert R Meyer,
April 16, 2010
lec 10F.2
Inclusion-Exclusion
|A B| = ?
A
B
A
What if not disjoint?
Albert R Meyer,
April 16, 2010
What if not disjoint?
lec 10F.3
Inclusion-Exclusion (3 Sets)
Albert R Meyer,
April 16, 2010
April 16, 2010
lec 10F.4
A town has n clubs.
Each club Si has a secretary
Mi who knows if person x is
a club member:
Mi(x) = 1 if x in Si,
= 0 if x not in Si.
B
C
Albert R Meyer,
Incl-Excl Formula: Proof
|ABC| =
|A|+|B|+|C|
B| – |A
C| – |B
C|
– |A
B
C|
+ |A
A
B
lec 10F.8
Albert R Meyer,
April 16, 2010
lec 10F.10
1
Incl-Excl Formula: Proof
Incl-Excl Formula: Proof
A = xpeople M A (x)
So
Mi(x)Mj(x) is sec’y for SiSj, so
S i S j = x M i (x) M j (x)
S i S j S k = x M i (x)M j (x)M k (x)
Let D ::= S1S2Sn
sec’y MD(x)=0 iff Mi(x)=0 for
all n clubs. So
1-MD(x) =
(1-M1(x))(1-M2(x))(1-Mn(x))
etc.
Albert R Meyer,
April 16, 2010
lec 10F.11
Incl-Excl Formula: Proof
MD(x) = i Mi(x)
- i<jMi(x)Mj(x)
+ i<j<k Mi(x)Mj(x)Mk(x)
April 16, 2010
lec 10F.13
|D| = i |Si|
- i<jMi(x)Mj(x)
+ i<j<k Mi(x)Mj(x)Mk(x)
Albert R Meyer,
April 16, 2010
lec 10F.14
1-MD(x) = (1-M1(x))(1-M2(x))(1-Mn(x))
now sum both sides over x
|D| = i |Si|
- i<j|SiSj|
+ i<j<k Mi(x)Mj(x)Mk(x)
(-1)n+1 M1(x)M2(x)Mn(x)
April 16, 2010
MD(x) = i Mi(x)
- i<jMi(x)Mj(x)
+ i<j<k Mi(x)Mj(x)Mk(x)
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))(1-Mn(x))
now sum both sides over x
Albert R Meyer,
1-MD(x) = (1-M1(x))(1-M2(x))(1-Mn(x))
now sum both sides over x
+ (-1)n+1 M1(x)M2(x)Mn(x)
Incl-Excl Formula: Proof
+
lec 10F.12
(-1)n+1 M1(x)M2(x)Mn(x)
Albert R Meyer,
April 16, 2010
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))(1-Mn(x))
so...
+
Albert R Meyer,
+ (-1)n+1 M1(x)M2(x)Mn(x)
lec 10F.15
Albert R Meyer,
April 16, 2010
lec 10F.16
2
Incl-Excl Formula: Proof
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))(1-Mn(x))
now sum both sides over x
|D| = i |Si|
- i<j|SiSj|
+ i<j<k |SiSjSk|
+ (-1)n+1 |S1S2
Sn|
(-1)n+1 M1(x)M2(x)Mn(x)
Albert R Meyer,
April 16, 2010
|D| = i |Si|
- i<j|SiSj|
+ i<j<k |SiSjSk)|
+
1-MD(x) = (1-M1(x))(1-M2(x))(1-Mn(x))
now sum both sides over x
lec 10F.17
Albert R Meyer,
April 16, 2010
lec 10F.18
Team Problems
Incl-Excl Formula: Proof
Problems
1—3
|D| = i |Si|
- i<j|SiSj|
+ i<j<k |SiSjSk|
+ (-1)n+1 |S1S2
Sn|
Albert R Meyer,
April 16, 2010
lec 10F.19
Albert R Meyer,
April 16, 2010
lec 10F.22
3
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6.042J / 18.062J Mathematics for Computer Science
Spring 2010
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