3Sets and CountingThis section is my notes on Chapter 6 of [1].3.1SetsDefinition 16.Asetis a collection of items, which are referred to as theelementsof the set. We use acapital a letter to name a set and braces to enclose its elements.•x∈Ameans thatxis an element of the setA.•Ifxis not an element ofA, we writex /∈A.•B=Ameans thatAandBhave the same elements. The order in which the elements are listed doesnot matter, i.e.{1,2,3}={2,3,1}.•B⊆Ameans thatBis asubsetofA, i.e. every element ofBis also an element ofA. Note thatsayingA=Bis equivalent to saying bothA⊆BandB⊆A!•IfBis a subset ofAthat is not equal toA, then we we writeB⊂Aand say thatBis apropersubsetofA.•Denote by∅theempty set, which contains no elements and is a subset of every set.•Afinite setis a set which has finitely many elements.•Aninfinite setis a set which does not have finitely many elements.Example 29.Consider the setW={Amazon, eBay, Apple}. We have:•Amazon∈W•Microsoft/∈W• {eBay, Apple, Amazon} ⊆WandW⊆ {eBay, Apple, Amazon}• {eBay, Apple} ⊂Wa proper subset.• {Amazon, eBay, Apple}={Apple, eBay, Amazon}•∅⊆Wand∅⊂W.•Wis a finite set.Example 30.Consider the setN=N={1,2,3,4, . . .}, thenatural numbers. One has:•2∈N•1009∈N•12/∈N•π /∈N• {1,2,3}and{1,3}are both finite proper subsets ofN.• {1,3} ⊂ {1,2,3}•Nis an infinite set.12