MATH 74 FINAL REVIEW
(1)
Basic Set Theory, Logic and Functions
: set operations (intersection, complement, power set,
Cartesian product, etc.), predicate logic, truth tables, basic properties of functions (injectivity, com
position, inverse image, etc.)
(2)
Cardinality
: Pigeonhole principle, countable and uncountable sets.
(3)
Metric Spaces, Continuity and Basic Topology
: Open and closed sets, open balls, diﬀerent
equivalent notions of continuity, limit points, closure of sets, homeomorphisms.
(4)
Sequences
: Cauchy sequences, complete metric spaces, limits and continuity, supremum, inﬁmum.
(5)
Compactness
: bounded, totally bounded, open covers, continuous image of compact sets are com
pact, HeineBorel Theorem, extreme value theorem.
(6)
Connectedness
: connected subsets of
R
, continuous image of connected sets are connected, inter
mediate value theorem.
(7)
Groups and Homomorphisms
: Deﬁnitions, elementary properties of groups and homomorphisms
(e.g. uniqueness of identities and
φ
(
x

1
) =
φ
(
x
)

1
), kernels of homomorphisms, abelian groups,
order of a group.
(8)
Examples of Groups
:
Z
/n
Z
and general cyclic groups,
D
n
,
S
n
,
GL
(
n
),
SL
(
n
),
O
(
n
),
SO
(
n
)
(9)
Subgroups and Cosets
: Lemma for determining if a subset is a subgroup, multiplication modulo
a subgroup
H
, left and right cosets, normal subgroups.
(10)
Quotient Groups
: Quotient groups, coset multiplication, equivalence between surjective homo
morphisms and normal subgroups, ﬁrst isomorphism theorem.
(11)
Equivalence Relations
: Deﬁnition, equivalence classes, Theorem: a partition of a set
X
into
mutually disjoint subsets is the same as an equivalence relation (statement only).
(12)