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Lecture 1: Introductory set theory, the set of natural numbers and
proof by induction.
1.
Basic set theory
•
Deﬁnitions of subset, set equality, union, intersection, disjointedness, empty
set, power set, cartesian product.
Ex: Prove that
C
\
(
A
∪
¯
B
) =
A
∪
(
¯
B
∪
¯
C
)
.
Note: Any statement “
∀
x
∈ ∅
...” is automatically true (i.e. a tautology).
•
Set notation and indexing families.
Ex: Let
A
k
= [0
,
2

1
/k
]. Find
S
∞
k
=1
A
k
.
Ex: Let
B
k
= (

1
/k,
1
/k
). Find
T
∞
k
=1
B
k
.
•
Deﬁnition of function, domain, range, image, preimage.
Ex:
f
(
f

1
(
D
))
⊆
D
,
f
(
C
1
∪
C
2
) =
f
(
C
1
)
∪
f
(
C
2
). What about
∩
?
•
Surjective, injective, and bijective functions. Deﬁnition of the inverse, and
when it is a function.
Ex:
f
(
x
) =
e
x
,
f
(
x
) =
x
(
x

1)(
x
+ 1),
f
(
x
) =
x
3
.
•
Deﬁnitions of countable and uncountable.
Ex:
Z
and
Q
are countable.
Ex:
R
is uncountable.
2.
The natural numbers and proof by induction
•
Deﬁnition of the set
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