ebookmass.com SCHWESERNOTES™ 2024 FRM® PART I BOOK 1: FOUNDATIONS OF RISK MANAGEMENT Kaplan Schweser
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A proposition,usuallydenotedby p,isadeclarativesentencethatiseither trueorfalse,butnotboth.
Definition1.2NegationofaProposition
If p isaproposition, ¬p isthe negation of p.Theproposition p istrueif andonlyifthenegation ¬p isfalse.
Fromtwopropositions p and q,wecanapplylogicaloperatorsandobtaina compoundproposition.
Definition1.3ConjunctionofPropositions
If p and q arepropositions, p ∧ q isthe conjunction of p and q,readas"p and q".Theproposition p ∧ q istrueifandonlyifboth p and q aretrue.
Definition1.4DisjunctionofPropositions
If p and q arepropositions, p ∨ q isthe disjunction of p and q,readas"p or q".Theproposition p ∨ q istrueifandonlyifeither p istrueor q istrue.
Definition1.5ImplicationofPropositions
If p and q arepropositions,theproposition p → q isreadas"p implies q". Itisfalseifandonlyif p istruebut q isfalse.
p → q canalsobereadas"if p then q or"p onlyif q".Inmathematics,we usuallywrite p =⇒ q insteadof p → q
Definition1.6DoubleImplication
If p and q arepropositions,theproposition p ←→ q isreadas"p ifand onlyif q".Itistheconjunctionof p → q and q → p.Hence,itistrueifand onlyifboth p and q aretrue,orboth p and q arefalse.
Thestament“p ifandonlyif q”isoftenexpressedas p ⇐⇒ q Twocompoundpropositions p and q aresaidtobelogicallyequivalent,denoted by p ≡ q,providedthat p istrueifandonlyif q istrue.
Whenthedomainsfor x and y areboththesetofrealnumbers,thefirststatement istrue,whilethesecondstatementisfalse.
Foraset A,weusethenotation x ∈ A todenote x isanelementoftheset A; andthenotation x/ ∈ A todenote x isnotanelementof A
Definition1.7EqualSets
Twosets A and B areequaliftheyhavethesameelements.Inlogical expression, A = B ifandonlyif
Definition1.8Subset
If A and B aresets,wesaythat A isa subset of B,denotedby A ⊂ B, ifeveryelementof A isanelementof B.Inlogicalexpression, A ⊂ B meansthat
x ∈ A =⇒ x ∈ B.
1. ¬ (∀xP (x)) ≡∃x ¬P (x)
2. ¬ (∃xP (x)) ≡∀x ¬P (x)
Chapter1.TheRealNumbers4
When A isasubsetof B,wewillalsosaythat A iscontainedin B,or B contains A.
Wesaythat A isa propersubset of B if A isasubsetof B and A = B.In sometextbooks,thesymbol"⊆"isusedtodenotesubset,andthesymbol"⊂" isreservedforpropersubset.Inthisbook,wewillnotmakesuchadistinction. Wheneverwewrite A ⊂ B,itmeans A isasubsetof B,notnecessaryaproper subset.
If A and B aresets,the union of A and B istheset A ∪ B whichcontains allelementsthatareeitherin A orin B.Inlogicalexpression,
Definition1.10IntersectionofSets
If A and B aresets,the intersection of A and B istheset A ∩ B which containsallelementsthatareinboth A and B.Inlogicalexpression,
Definition1.11DifferenceofSets
If A and B aresets,thedifferenceof A and B istheset A \ B which containsallelementsthatarein A andnotin B.Inlogicalexpression,
Definition1.12ComplementofaSet
If A isasetthatiscontainedinauniversalset U ,the complement of A in U istheset AC whichcontainsallelementsthatarein U butnotin A.In logicalexpression,
Chapter1.TheRealNumbers5
Sinceauniversalsetcanvaryfromcontexttocontext,wewillusuallyavoid usingthenotation AC anduse U \ A insteadforthecomplementof A in U .The advantageofusingthenotation AC isthatDeMorgan’slawtakesamoresuccint form.
Proposition1.4DeMorgan’sLawforSets
If A and B aresetsinauniversalset U ,and AC and BC aretheir complementsin U ,then
1. (A ∪ B)C = AC ∩ BC
2. (A ∩ B)C = AC ∪ BC
Definition1.13Functions
When A and B aresets,a function f from A to B,denotedby f : A → B, isacorrespondencethatassignseveryelementof A auniqueelementin B.
If a isin A,the image of a underthefunction f isdenotedby f (a),andit isanelementof B.
A iscalledthe domain of f ,and B iscalledthe codomain of f
Definition1.14ImageofaSet
If f : A → B isafunctionand C isasubsetof A,theimageof C under f istheset
f (C)= {f (c) | c ∈ C}
f (A) iscalledtherangeof f
Definition1.15PreimageofaSet
If f : A → B isafunctionand D isasubsetof B,thepreimageof D under f istheset
f 1(D)= {a ∈ A | f (a) ∈ D} .
Noticethat f 1(D) isanotation,itdoesnotmeanthatthefunction f hasan inverse.
Wesaythatafunction f : A → B isan injection,orthefunction f : A → B is injective,orthefunction f : A → B is one-to-one,ifnopairof distinctelementsof A aremappedtothesameelementof B.Namely,
Wesaythatafunction f : A → B isa surjection,orthefunction f : A → B is surjective,orthefunction f : A → B is onto,ifeveryelementof B istheimageofsomeelementin A.Namely,
Equivalently, f : A → B issurjectiveiftherangeof f is B.Namely, f (A)= B
Definition1.18Bijection
Wesaythatafunction f : A → B isa bijection,orthefunction f : A → B isbijective,ifitisbothinjectiveandsurjective.
Abijectionisalsocalleda one-to-onecorrespondence. Finally,wewouldliketomakearemarkaboutsomenotations.If f : A → B isafunctionwithdomain A,and C isasubsetof A,therestrictionof f to C is thefunction f |C : C → B definedby f |C (c)= f (c) forall c ∈ C.Whenno confusionarises,wewilloftendenotethisfunctionsimplyas f : C → B
Thesetof naturalnumbers N isthesetthatcontainsthecountingnumbers, 1,2,3 ,whicharealsocalledpositiveintegers.
N isaninductiveset.Thenumber1isthesmallestelementofthisset.If n is anaturalnumber,then n +1 isalsoanaturalnumber. Thenumber0correspondstonothing.
Foreverypositiveinteger n, n isanumberwhichproduces0whenaddsto n.Thisnumber n iscalledthenegativeof n,ortheadditiveinverseof n 1, 2, 3, ...,arecallednegativeintegers.
Definition1.20Integers
Thesetof integers Z isthesetthatcontainsallpositiveintegers,negative integersand0.
Thesetofcomplexnumbers C isthesetthatcontainsallnumbersofthe form a + ib,where a and b arerealnumbers,and i isthepurelyimaginary numbersuchthat i2 = 1.Itcontainsthesetofrealnumbers R asasubset. Additionandmultiplicationcanbeextendedtothesetofcomplexnumbers.These twooperationsoncomplexnumbersalsosatisfyallthepropertieslistedabove. Nevertheless,weshallfocusonthesetofrealnumbersinthiscourse.
Givenarealnumber x,the absolutevalue of x,denotedby |x|,isdefined tobethenonnegativenumber
Inparticular, |− x
Forexample, |
Theabsolutevalue |x| canbeinterpretedasthedistancebetweenthenumber x andthenumber 0 onthenumberline.Foranytworealnumbers x and y, |x y| isthedistancebetween x and y.Hence,theabsolutevaluecanbeusedtoexpress aninterval.
Chapter1.TheRealNumbers12
IntervalsDefinedbyAbsoluteValues
Let a bearealnumber.
1. If r isapositivenumber, |x a| <r ⇐⇒−r<x a<r ⇐⇒ x ∈ (a r,a + r).
|x 5|≤ 2 implies 3 ≤ x ≤ 7.Thismeansthat x ispositive.The inequality x ≥ 3 thenimpliesthat x2 ≥ 9,andtheinequality x ≤ 7 implies that x2 ≤ 49.Therefore, 9 ≤ x 2 ≤ 49.
Finally,wehavetheusefulCauchy’sinequality.
Chapter1.TheRealNumbers14
Proposition1.9Cauchy’sInequality
Thisisjustaconsequenceof
AnimmediateconsequenceofCauchy’sinequalityisthearithmeticmeangeometricmeaninequality.Foranynonnegativenumbers a and b,thegeometric meanof a and b is √ab,andthearithmeticmeanis
Proposition1.10
Chapter1.TheRealNumbers15
Exercises1.2
Question1
Useinductiontoshowthatforanypositiveinteger n, n! ≥ 2n 1
Let S beanonemptysubsetofrealnumbersthatisboundedabove,and let US bethesetofupperboundsof S.Then US isanonemptysetthatis boundedbelow.If US hasasmallestelement u,wesaythat u isthe least upperbound or supremum of S,anddenoteitby u =sup S.
Let S beanonemptysubsetofrealnumbers.Then S hasamaximumifand onlyif S isboundedaboveand sup S isin S
Onenaturalquestiontoaskis,if S isanonemptysubsetofrealnumbersthat isboundedabove,does S necessarilyhavealeastupperbound.Thecompleteness axiomassertsthatthisistrue.
CompletenessAxiom
If S isanonemptysubsetofrealnumbersthatisboundedabove,then S hasaleastupperbound.
