Review Guide for Math S101 final exam – Tuesday August 16^th

The final exam for this class will be on Tuesday from 1:30 to 4:30pm in room 221 in the Science Center. The exam will be cumulative meaning that questions can come from any of the topics covered this summer, not just those we’ve studied recently. On the test you can expect to see questions similar in style to the ones you’ve already seen on the first two tests. There will probably be around 8 or 9 problems in total. Problem types might include “prove the following” or “spot the error in the following proof” or “make a conjecture based on the following information.”

To get ready you should again go through your class notes and through the readings from the Wolf textbook and the Notes. Make an outline of what has been covered so far by listing out the various topics and providing an example or two of each key point. It might very well be the case that several of the test questions look quite similar to previous homework questions, so it will be a good idea to review all of the homework assignments that you’ve completed, as well as to check through the answers that Emily has posted on the website.

As you go through the material we’ve covered this summer think hard about the strategy behind each proof we’ve seen in class or in the textbooks. Memorize the different techniques we’ve seen – just being familiar with a proof is often not enough. I’ve been frustrated at times remembering that I’d seen a proof of a particular result, but not being able to remember the particular trick that made a proof possible.

Topics covered on this final exam will include:

Everything from the review list for the First Midterm (which I’ve copied in below):

• Types of proofs/problem solving techniques
• Principle of Mathematical Induction
• prove a base case (usually for n = 0 or n = 1)
• prove the induction step (case n implies case n+1)
• Proof by contradiction
• assume the negation of the statement to be proved
• derive a contradiction
• Proof by consideration of possible cases
• similar approaches to different cases might work
• check parity (when there are two choices such as odds/evens)
• Pigeon Hole Principle
• Notational issues
• Review notation we’ve used so far:
• R, Q, Z, N etc for number systems
• Use of quantifiers such as $ for “there exists”
• bracket notation, set builder notation for set theory
• Number systems
• Knowledge of the 11 possible properties for a number system
• two operations, each with closure, identity element, associativity, inverses, commutativity
• distributive property that bringes the two operations together
• Know some basic examples such as the integers (Z has all 11 properties except no multiplicative inverses)
• Basic proofs using these properties (e.g. that M x 0 = 0 in any number system)
• Propositional Logic
• Knowledge of five connectives and their truth tables
• Know how to construct truth tables to evaluate logic statements
• Set Theory
• Know the basic operations – intersection, union, complement
• Know how to construct and use Venn diagrams
• Review any basic proofs involving sets (e.g. if A Ì B then A U B = B, and vice versa - if A U B = B then A Ì B (lemma on page 18 of Notes )).
• Closure Operators
• Definition of a closure operator
• know the four closure operator axioms
• Examples of closure operators
• review examples of closure operators we’ve seen as well as operators that don’t satisfy the closure axioms

And now here are more topics covered since the first midterm:

• Closed and Open sets (chapter 2 in the Notes)
• Know the results covered in Theorem 2.7 (the Closed Set Theorem)
• Know what Theorem 2.7 implies about open sets
• Infinite unions of open sets are still open
• Intersections of any finite number of open sets are open
• Know the results covered in sections 2.2, 2.3 and 2.4 as they are used for Theorem 2.7
• Euclidean Closure Operator (chapter 3 in the Notes)
• Know the definition of a (distance) metric
• satisfies nonnegativity, symmetry and triangle inequality (definition 3.3)
• know some basic examples (euclidean, taxicab metric)
• Review the proof of the fact that the Euclidean Closure operator satisfies the four properties of a closure operator
• Review the definition of an open ball in Rⁿ and the alternate definition of the Euclidean Closure operator using the open ball definition.
• Set theory – countability
• review basic definitions – a set is called countable if it is either finite or if it can be put in one-to-one correspondence with the natural numbers (i.e. its elements can be listed out)
• review proofs of the countability of the rationals, the uncountability of the real numbers, the uncountability of the power set of Z
• Review the proof of the irrationality of the square root of two and other non-square integers
• Also be sure to review the games we went through during the summer, studying the winning strategies – there will likely be a question involving a game winning strategy on the final

Topics that you can skip as you prepare for the final:

• There will be no questions on the Cantor set
• There will be no questions involving the topological proof of the infinity of primes
• There will be no questions from Chapter 4 on connected sets