Here is an example. Suppose we have 100 students, of which 50 take
French, 30 take Spanish, and 20 take both. How many take at least one of
the two languages?
30 = 50

20 take just French, 10 = 30

20 take just Spanish, and 20 take
both, so 70 = 30 + 10 + 20 take at least one.
Proposition 1.3
If
A
⊂
B
, then
P
(
A
)
≤
P
(
B
)
.
Proof.
P
(
B
) =
P
(
A
) +
P
(
B
∩
A
c
)
≥
P
(
A
)
.
Proposition 1.4
Suppose
A
i
↑
A
. This means
A
1
⊂
A
2
⊂ ···
and
A
=
∪
∞
i
=1
A
i
. Then
P
(
A
) = lim
i
→∞
P
(
A
i
)
.
Proof.
Let
B
1
=
A
1
,
B
2
=
A
2
∩
A
c
1
,
B
3
=
A
3
∩
A
c
2
, and so on. Then the
B
i
are disjoint, their union is
A
, and the union of the ﬁrst
n
of them is
A
n
. So
by the deﬁnition of inﬁnite sum,
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 ansan

Click to edit the document details