Exam II will be in the usual classroom on Thursday, March 23, 2006. Calculators or other mechanical assistance are not needed and are not to be used. Blank paper will be provided, so all you will need is something to write with.

Two of the questions will be similar to questions that appeared on Exam I. Be sure that you understand well how to do all the problems on Exam I, so that you can easily do the similar questions.

Apart from the repeat questions, the exam will cover sets and functions. It will have definitions, problems similar to those on the homework, and questions related to the lectures. On this exam, as on all exams in this course, “Prove” means “Give an argument”, not “Tell why”.

The scope of mathematics is the precisely definable, and the underlying definitions form the foundation of the theoretical ideas. One can know the definitions without knowing the mathematics, but one cannot know the mathematics without knowing the definitions. The exam will have a lot of questions asking you simply to give definitions, and this means a precise definition, using logic notation and/or set notation when appropriate, like the definitions we examined in class. Knowing a definition means being able to apply it in different contexts, adapting the notation as necessary. For example, the definition of “$A\; \subseteq \; B$” is “ $\forall \; x,\; x\; \in \; A\; \Rightarrow \; x\; \in \; B$ (your browser might not display all of these logic symbols), but if asked to define “$y\; \subseteq \; x$”, one needs to adapt it as something like “ $\forall \; t,\; t\; \in \; y\; \Rightarrow \; t\; \in \; x$.

The following will definitely be covered, although the exam not necessarily limited to these topics: intersection, union, and equality of sets, subsets, Cartesian product of sets, power sets, countable infinite sets, functions (including all basic definitions, such as domain, codomain, preimage, composition, equality of functions, inverse function, graph, etc.), surjective, injective, and bijective functions (definitions, proofs that specific functions are or are not surjective or bijective, examples).
One should be familiar with the standard notations ( N, Z, Q, R, C ) that we use for the sets of natural numbers, integers, rational numbers, real numbers, and complex numbers, and with interval notation (a,b), [a,b), etc.

The following do not appear on this exam (which is not to say that they are unimportant): the concept of multiplicity, Venn diagrams, Rolle's theorem, tricky membership or subset relationships involving the empty set, vacuously true statements, n-tuples other than pairs, preimage of a subset.

Exams from last fall's class can be found on its course page (links to my previous course pages appear on the course pages page). That course had 50-minute classes, so our exam will be somewhat longer. Different classes are different, and the exams will different, but in general it's a good technique to rehearse an exam using others from previous classes.