ECON6003W1
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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 > 1”isvacuouslytrue.
v. ¬( ⇒ ): ¬ ⇒ ¬; ¬( ⟺ ): ¬ ⟺ ¬
vi. DeMorgan’slaw: ¬( ∧ ) ≡ ¬ ∨ ¬; ¬( ∨ ) ≡ ¬ ∧ ¬
|
|
∧ |
∨
|
¬ 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 objectswhicharecalled“elements”
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 ×
2022-04-12