MATH 74 FINAL REVIEW
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.)
: Pigeonhole principle, countable and uncountable sets.
Metric Spaces, Continuity and Basic Topology
: Open and closed sets, open balls, diﬀerent
equivalent notions of continuity, limit points, closure of sets, homeomorphisms.
: Cauchy sequences, complete metric spaces, limits and continuity, supremum, inﬁmum.
: bounded, totally bounded, open covers, continuous image of compact sets are com-
pact, Heine-Borel Theorem, extreme value theorem.
: connected subsets of
, continuous image of connected sets are connected, inter-
mediate value theorem.
Groups and Homomorphisms
: Deﬁnitions, elementary properties of groups and homomorphisms
(e.g. uniqueness of identities and
), kernels of homomorphisms, abelian groups,
order of a group.
Examples of Groups
and general cyclic groups,
Subgroups and Cosets
: Lemma for determining if a subset is a subgroup, multiplication modulo
, left and right cosets, normal subgroups.
: Quotient groups, coset multiplication, equivalence between surjective homo-
morphisms and normal subgroups, ﬁrst isomorphism theorem.
: Deﬁnition, equivalence classes, Theorem: a partition of a set
mutually disjoint subsets is the same as an equivalence relation (statement only).