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:  ¬( ∧ ) ≡ ¬ ∨ ¬;  ¬( ∨ ) ≡ ¬ ∧ ¬

2.   Proof ofaproposition:assumption+logicaloperation

a)    Constructive/deductiveproof

b)   Contraposition:conversionofapropositionfromall  ¬ ⇒ ¬ to all ⇒

c)    Bycontradiction(mostcommonone) fromall ⇒ ¬ isfalseto all ⇒

-

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 ∆ = (\) ∪ (\) = ( ∪ )\( ∩ )