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Final Review Sheets

# Final Review Sheets - MATH 74 FINAL REVIEW(1 Basic Set...

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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, Heine-Borel 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) Lagrange’s Theorem : Both the statement and proof of the theorem, theorem applied to the order of group elements, (13) Conjugacy Classes and Cauchy’s Theorem

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Final Review Sheets - MATH 74 FINAL REVIEW(1 Basic Set...

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