MA361 Final Exam Skill Inventory

Definitions and Terminology

• upper bound for a set
• least upper bound or supremum
• greatest lower bound or infimum
• 1-1 function
• onto function
• the Axiom of Completness
• the Nested Interval Property
• the Monotone Convergence Theorem
• the Bolzano-Weierstrass Theorem
• the Cauchy Criterion
• Cauchy sequence
• 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
• equivalent sets (sets having the same cardinality)
• open set
• closed set
• limit point
• isolated point
• the closure of a set
• compact set
• bounded set
• open cover
• finite subcover
• perfect set
• separated sets
• disconnected set
• connected set
• F-sigma set
• G-delta set

One column will contain the term or the name of a theorem.

In Mathematics, the choice of whether an "if and only if" statement is considered to be a theorem or a definition is somewhat arbitrary.

For example, the author defines a compact set as a set K with the property that every sequence in K has a subsequence that converges to a limit in K.

Later, he proved that a set K is compact if and only if every open cover of K has a finite subcover.

In many Real Analysis texts, the open cover theorem is used as the definition of a compact set, and the subsequence property is introduced as a theorem.

Consequently, the item that matches a term may be either the definition given, or some condition that was shown to be equivalent in a theorem. For example,

a.  compact set	( ) A set A with the property that every sequence in A has a subsequence that converges to limit in A
	( ) A set having the property that every open cover has a finite subcover

In this example, the term matches both statements, so the correct answer is

(a) A set A with the property that every sequence in A has a subsequence that converges to limit in A

(a) A set having the property that every open cover has a finite subcover

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).

Candidates for proofs you have seen are listed in the attached document.

Candidates for proofs you have not seen before:

• 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 open (or closed)
• Prove that a given set is compact (or not compact)
• Prove that a given set is connected (or not connected)
• Prove that a given recursive sequence (that lends itself to induction proofs) converges, and/or find its limit.