Math 310B, Fall 2008
Final Topics & Overview
Final Exam: Monday, December 8, 2:30-4:20, MEB 235
You may bring two 8.5x11 sheets of notes (either one or two-sided, but stapled together) and a
calculator.
The final is comprehensive, including chapters 1-14, 19-20, and 24 of the text. The material which was
not covered by the midterm will get somewhat more weight.
Study: Your class notes, the text (examples, proofs, assigned end-of-chapter problems), and the
collected homework problems.
More practice problems are posted (partial solutions will be posted later). Bring questions to reviews or
office hours.
Final exam questions may include computational questions (for instance: counting how many ways you
can do a certain task), definitions, multiple choice or Yes/No questions, short questions like “give the
converse of the following statement”, finding errors in an argument, short proofs, a couple “serious”
proofs.
Main topics per section:
I. Recall the old (midterm) material:
1. The Language of Mathematics: Statements, Connectives (and, or, not).
2. Implications.
3. Direct Proofs and proof by cases.
4. Proof by contradiction.
5. Induction
6. Sets: elements, ways to define/notation, subsets, empty set, operations on sets (union,
intersection, difference), power set of a set, complement of a set.
Thm 6.3.4: results and how to prove them.
Know how to prove a set is a subset of another, or that two sets are equal.
Understand difference between an element and a subset.
7. Quantifiers: universal and existential.
Understand what they are and how to use them. Understand combinations of more than one
quantifier, and negations of such. How to prove and disprove statements involving quantifiers.
Cartesian product: what is it?
8. Functions
Definitions & understand: functions, domain, codomain, image, graph, composition of functions.
Be able to come up with examples!
How to formally prove that the limit of a sequence is (or is not) equal to a number.
9. Functions: Injections, Surjections and Bijections.
Understand, be able to prove or disprove, be able to give examples.
Inverse of a function: what it is, its domain/codomain, and how to find it.
(You may skip: Functions on subsets ( 𝑓
and 𝑓
).)
