Chapter 1 Skill Inventory

Definitions and Terminology

To prepare for exam 2, you should be familiar with the following terms:

an upper (or lower) bound for a set
a set that is bounded above (or below)
the least upper bound or supremum
the greatest lower bound or infimum
the maximum (or minimum) of a set
a 1-1 function
an onto function
the Axiom of Completness
the Nested Interval Property
a countable (uncountable) set
the set of natural numbers
the set of integers
the set of rationals
the set of irrationals
the set of algebraic numbers
the set of transcendental numbers
the power set
the cardinality of a set
equivalent sets (sets having the same cardinality)
an ordering (p. 246 in the text)
an ordered field (p. 246 in the text)

Proofs

Exam 2 will contain a number of proofs.

Due to the nature of proofs, there are obvious limitations to what you can be expected to do in a single class period.

The proofs on the exam will fall into two categories:

Proofs you have seen before.
Proofs you have not seen before (but are reaonably similar to proofs you have seen).

Proofs in the latter category will be relatively easy.
Proofs in the former category may be more difficult, but you will be able to prepare for them in advance.

Candidates for proofs you have seen:

Proof that the square root of 2 is irrational
Proof of the Nested Interval Property
Proof of the Archimedean Property
Cantor's Diagonalization Method to prove that (0,1) is uncountable
Proof that a nonempty finite set must contain its supremum

Candidates for proofs you have not seen before:

A simple induction proof
Given set, prove that a certain number is (or is not) the supremum of the set (using Lemma 1.3.7)
Prove that a given set is countable
Prove that a given function is 1-1
Prove that two sets are equal
Prove that two sets are equivalent (have the same cardinality)