[PDF] Lecture 16 : Definitions, theorems, proofs Meanings Examples

28012_6AxiomNotes.pdf Math 299Lecture 16 : Denitions, theorems, proofsMeanings Denition: an explanation of the mathematical meaning of a word. Theorem:A statemen tthat has b eenpro vento b etrue . Proposition: A less important but nonetheless interestingtrue statemen t. Lemma:A true stateme ntused in p rovingoth ertrue stateme nts(that is, a less important theorem that is helpful in the proof of other results). Corollary:A true statmen tthat is a simple deduction from a theorem or prop osition. Proof: The explanation of why a statement is true. Conjecture: A statement believed to be true, but for which we have no proof. (a statement that is being prop osed to b ea true statem ent). Axiom: A basic assumption about a mathematical situation. (a statement we assume to be true).

Examples

Denition 6.1: Astatementis a sentence that is either true or false{but not both. ([H], Page 53). Theorem 10.1:N, considered as a subset ofR, is not bounded above. ([B], Page 96).
Corollary 10.2:Zis not bounded above. ([B], Page 96). Proposition 10.4: For each" >0, there existsn2Nsuch that1n < ". ([B], Page 96).
Lemma:Lemmas are considered to be less important than propositions. But the distinction between categories is rather blurred. There is no formal distinction among a lemma, a proposition, and a theorem. Axioms: Ifmandnare integers, thenm+n=n+m. (Read [B] Page 4.) Conjecture: Mathematicians are making, testing and rening conjectures as they do their research.

Group Axioms

AGroupis a setGtogether with an operation #, for which the following axioms are satised. A

1:Closure:8a;b2G; a#b2G

A

2:Associativity:8a;b;c2G;(a#b)#c=a#(b#c)

A

3:Identity element:9e2Gsuch that8a2G; a#e=e#a=a

A

4:Inverse element:8a2G;9b2Gsuch thata#b=b#a=e

1.

Is Nwith + a group?

2.

Is Zwith + a group?

3. Do the axioms imply that i fGis a group anda;b2Gthena#b=b#a? 4. Can y ougiv ean example of a group (all axioms A1A4are satised) whose elements do not commute with each other? . Group TheoremsThereom:The identity element is unique. proof:Thereom:For every elementa2Gthere exists auniqueinverse. proof:

Axiomatic system 1:

Undened terms:member, committee

A

1:Every committee is a collection of at least two members.

A

2:Every member is on at least one committee.

1.

Find t wodieren tm odelsfor this set of axioms.

2. Discuss ho wit can b emade categorical(there is a one-to-one correspondence be- tween the elements in the model that preserves their relationship). Axiomatic system 2:Denition:A line`intersectsa linemif there is a pointAthat lies on both`and m.Undened terms:point, line A

1:Every line is a set of at least two points.

A

2:Each two lines intersect in a unique point.

A

3:There are precisely three lines.

Find two dierent models for this set of axioms.

Axiomatic system 3:

Undened terms:point, line, lie onDenition:A line`passes throughpointsAandBifAandBlie on`.Denition:A line`intersectsa linemif there is a pointAthat lies on both`and

m. A

1:There are exactly ve points.

A

2:Exactly two points lie on each line.

A

3:At most one line passes through any two points.

A

4:There are exactly three lines.

Find two dierent models for this set of axioms.Theorem:At least one pair of lines intersect. Conjecture:There is a point that doesn't lie on any line.