Ma185 Exercise Set 1
-
Upload
riemann-soliven -
Category
Documents
-
view
222 -
download
0
Transcript of Ma185 Exercise Set 1
-
7/21/2019 Ma185 Exercise Set 1
1/2
16 Chap. i Rules of the Road AxiomaticSystems
investigation shows that Axioms2 through 4 are all correct' Therefore'
Ax-
iom 1 is independent;;;;;;;"ining three'The reader is encouraged to flnd
models that demonsrr"t" irr" i"o"p"rrld"rr""of ,q^io-* 2 through 4. o
The property of completeness is also concerned with thesize of the axiom
set. wherea, i"d"pJ;:;:;;;;;;o 't'utour set or axioms was not too
large, completeness gti'"tlJ"hat our tt'o'"" *iotttsare sufficient in number
to prove or disprove ;;;;;-"it tnil."iit"s concerningour collection of
undefined terms' W" *v'tttut an axiom '"tis oisumcient sLe ot-complete \f
it
is impossible to add iil["it""J;;;;"tt u"a na"p"ttdentaxiom without
aaoneaaaitrH:i:lfi :t*ll"Tl""'n""1:'l:'^:::':,t;ffi Aswithconsis-
tency, the failure to fi;d a newconsist""t' ;;;;";dent"axiom does not
elimi-
nate the possibility ";il ;;;""ceand trr"r"rotJitun insufficient procedure by
which to p.ou" "o*pf"i"*"'W" tu"' t'o*"i"'' rr'" ttt" isomorphism of mod-
els to demon't'ut" tio'm of complete"""' ii;i;odelsof a given axiomatic
system are isomorpil;' ;; irt" **'ot utl*is said ro be categ'orical' Ttj
nrooertv of "ut"go'r"ii""" "u"ut 'tto*"
io iryptt completeness; the proof'
il,##i, i". ;;;?; ffi;;;" "t 1ry' discussion16
In the next section' we will illustratethe properties of axiomatic systems
ina geometric context'
EXERCISES 1.2
The Axiomatic Method
To answer Exercises 1 through 4' usethe axiomatic system outlined in
Exam-
ple 1.2.1'
1. Prove: A Fo cannot contain threedistinct Fe's'
2. Prove: There exists a set of two Fo's that contains all the Fe'sof the system'
3. Prove: For every set of two distinctFe's' every Fo in the system must
contain at least
one of them.
4. Prove: A1l three Fo's cannot containthe same Fe'
5. consider the following axiom set inwhich x's, y's, and "on" are the unde{ined
terms:
Axiom 1. There exist exactly flve x's'
Axiom2.Anytwodistinctx,shaveexactlyone}onbothofthem.
Axiom 3. Eachy is on exactly two x's'
How many y's are there in the system?Prove your result'
To answer Exercises 6 through 9' usethe axiom set
_ofcomp1etenessandcategorica1ness,seeJJ.ow?:9I'"*Founda-6 Fo, u detailed discusions and Fundamentar ,"iiio.zlr'i"ii-"*r,i6
(B.jr'# rw's-rENr Publishing co'' 1990)'
pp. 160-162.
in Exercise 5.
-
7/21/2019 Ma185 Exercise Set 1
2/2
Sec. 1.2 Axiomatic Systems and their Properties 17
6. Prove that an)r two l"s have at most one x on hoth'
7. Prove that not all -r's arc on the same y.
8. Prove that there exist exactly four y's on each 'r 'g. prove that for an] )1 and an] )1 not on that x1 there exist exactly two other distinct
l"s on 11 that do not contain any of the i's on ,)'1'
Models
l0.VerifythattheaxiomsinExamplel.2.2are..correct,'Statements.11. Verify that Axioms 1 ancl 3 in Example I.2.3 are "correct" statements and explain
why Axioms 2 and 4 are not correct.
12. Verify that the model inExample 1.2.5 is isomorphic to the modei in Example 1,2'2,
13. Devise a one-to-one correspondence between the undefined terms in the models in
E'xamplesl.2.5andl.2.6thatisanisomorphismandverifyyourresult.14. Devise another model that is isomorphic to the one in Example \.2.Z.Find a model
that is not isomorphic. if possible.
15. Devise a model for thc axiom system described in Exercise 5'
16. Consider an inflnite set of undeflned elements s ancl the undefined relation R that
satislics thc [ollou ing axioms:
Axiom l. 7f ,r, b e S and aRb, then a + b'Axirrm2. Itia. b, c e S.aRb,ancl bRc.thenaRc
(a) Show that an interpretation with s as the set of integers and aRb as "a is lessthan b" is a model for the syslem'
(lr) Would , as the set of integers and aRb interpreted as "a is gleater than b" alsobe a model?
(c)Are
the models in parts (a) and (b) isomorphic?
(d) Would S as the set of real numbers aIKl aRb interpreted as "a is less than b"be another model?
(e) ls the model in part (i1) isomorphic to the model in part (a)?
Properties of Axiomatic SYstems
L7. Devise two additional concrete models for the axiom system in Example 1'2.1' Are
these models isomorphic? Justify your result'
18. Devise two additional abstract models for the axiom system in E'xample 1'2'1'
lg.ArethemodelsinExercise16concreteorabstract?Explainyoulanswef.20. Explain why it is not possible to devise a conclete model of the real number system'
21. Demonstfare the independence of Axioms 2 through 4 in Example 1.2.1.
22. Devise two concrete models lor the axiom set in Exercise 5' Are these modelsisomorphic? Are they isomorphic to the model found in Exercise 15? Justify your
results.