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,
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 Spring '10
 ansan
 Logic, Harshad number, Zagreb

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