Review Guide for Math S101 first midterm – Tuesday July 19^th

This first test will be an in class test, and you will have the whole period to complete it. There will be a series of short answer questions (likely to be around 6 or 7 problems) and there will probably be no multiple choice (or essay) questions. Problem types might include “prove the following” or “spot the error in the following proof” or “make a conjecture based on the following information.” This will be different from a precalculus or calculus test in the sense that there won’t be any computation questions (i.e. “solve the following using the quadratic equation,” or “find the value of the following integral”). This means that as you get ready for this test you might find that you need to approach studying a bit differently from the way you’ve done this for previous math tests.

To get ready you should of course go through your class notes and through the readings that you’ve already done in the Wolf textbook and in 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. Be sure to review the several homework assignments that you’ve completed, as well as to check through the answers that Emily has posted on the website (they’re available by going to the assignments page).

One important aspect of your rereading is to make sure that you are “actively reading” while you go back through your notes and through the texts. This means that as you’re going through for a second (or third, fourth or fifth!) time you should rework examples – try doing them on your own. Be sure to write out key points on a separate piece of paper – make outlines as you read – so that you can follow the main line of the arguments. Pick out what’s most important and try not to get stuck in the small details, or on tangential issues as you try to figure out how various points fit together. Remember that just being familiar with the material isn’t the same as knowing how to use it and how to work problems based on it.

Remember also that one of the main reasons for having tests at all is to make sure that everyone spends time reviewing so that you can form a “big picture” of what’s happened so far. It’s an important part of the learning process, so take the time to do it well and try to get as much out of your studying as you can.

Topics covered on this first midterm will include:

• 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
• more pigeons than holes means 2 or more in one hole
• useful in situations when you’d like to show at least two of some object or number produce the same result (e.g. same remainder modulo a given number in the cases we worked with)
• 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 (potentially – they don’t all have to hold necessarily for all number systems)
• 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)
• Basic Propositional Logic
• Knowledge of five connectives and their truth tables
• Know how to construct truth tables to evaluate logic statements
• A Law of Propositional Logic is one in which the truth table shows all T values for all possible combinations of T/F for all of the propositional variables involved in the statement
• Propositional Logic Laws are also called tautologies
• Basic 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 and what they mean intuitively (e.g. points that are really close to points that are really close to A are themselves really close to A, e.g. KA = K(KA), for closure axiom 3)
• Examples of closure operators
• review examples of closure operators we’ve seen as well as operators that don’t satisfy the closure axioms

Notice that this list is not too short. When you first start to get ready for the test you might convince yourself that you don’t need to study because you’ve followed everything in class each day and it seemed relatively straightforward. However, this can be misleading as we’ve met now for nine sessions and have gone through a lot of different topics. It’s often quite easy to get confused in a test situation when one is presented with problems from a lot of different topics all at once, so do be sure to study to get ready!

In any case, don’t stress too much about the test – if it doesn’t go well the first time around, we can replace the score with your final test score, so just do your best to get ready and see you on Tuesday!