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ECON6003W1

1.   Proposition:

a)    Definition:astatementthatisunambiguouslyeithertrueorfalse(butnot both)inagivencontext

b)   Logicaloperation

i.        not:  ¬

ii.        and:  ∧

iii.        or:  ∨  (inclusiveinmathematics)

* exclusive“or”inmath:porqbutnotboth

iv.        ifp,thenq: ;pifandonlyif(iff)q:

* “ifpisF,then ” isvacuouslytrue.

* istrueexceptifpistrueandqisfalse.

Ex.“1 + 4 = 9 ⇒ 8 < 1”isvacuouslytrue;

1 + 2 = 3 8 < 1 isfalse; 1 + 1 = 3 8 > 1isvacuouslytrue.

v.        ¬():  ¬ ⇒   ¬;  ¬():  ¬ ⟺ ¬

vi.        DeMorganslaw:  ¬( ) ¬ ¬;  ¬( ) ¬ ¬

 ∧ ∨ ¬ or ¬ ⇒ ⟺ T T T T F T T T F F T T F F F T F T T T F F F F F T T T

2.   Proof ofaproposition:assumption+logicaloperation

a)    Constructive/deductiveproof

b)   Contraposition:conversionofapropositionfromall  ¬ ⇒ ¬ to all

c)    Bycontradiction(mostcommonone) fromall ⇒ ¬ isfalseto all

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d)   Byinduction

3.   Set:a well-specific collectionof distinct objectswhicharecalledelements

a)    Collection:nosequencewhile = (, , )  hassequence.

b)   Distinct:{a,a,b,c}isnotavalidset

c)    Description:

i.        Bylistingofallitselements: = {, , }

ii.        Bydescribingelements’commonproperty:

= { ∈ ℝ| ≥ 0}  or = {| = 10 ∗ , = 1,2,3,4}                                * The property should be a statement that is true or false.  (it’s not ambiguous)

d)   Well-specific: = { ∈ ℝ | ≥ 0}; = {| = 10 ∗ , = , , , }

e)    Symbol

i. : aisanelementofsetA;a belongstoA

ii. : disnotinA

iii. : AisasubsetofB;everyelementofAisinB();

* =

f)    Empty/nullset  ∅:theuniquesethavingnoelement

*  ∅  isthesubsetofanynon-emptyset(triviallyorvacuouslytrue)               The definitionofsubsetis .Then,if = ∅ , then thereisno , whichmeans “” if false. Accordingto “ifpisF,then ”isvacuouslytrue, “∅  isthesubsetofanynon-emptyset”isvacuously true.

g)    Universalset:collectionofalltheelementsunderconsideration.E.g.price

4.   Operation:(given , )

a)    Completement !  = { | }

b)   Union = { | }

c)    Intersection = { | }

d)   Setminus \ = { | } = !

e)    Symmetricdifference = (\) (\) = ( )\( )

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f)    CartesianProduct × = {(", # )|" , # }  (ex.)4.

* orderedpair  (", # )

* usuallydifferentfrom ×