# MA_114___Chapters_3_4_6_Notes (1).pdf - 3 Sets and Counting...

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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.xAmeans 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}.BAmeans thatBis asubsetofA, i.e. every element ofBis also an element ofA. Note thatsayingA=Bis equivalent to saying bothABandBA!IfBis a subset ofAthat is not equal toA, then we we writeBAand say thatBis apropersubsetofA.Denote bytheempty 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:AmazonWMicrosoft/W• {eBay, Apple, Amazon} ⊆WandW⊆ {eBay, Apple, Amazon}• {eBay, Apple} ⊂Wa proper subset.• {Amazon, eBay, Apple}={Apple, eBay, Amazon}WandW.Wis a finite set.Example 30.Consider the setN=N={1,2,3,4, . . .}, thenatural numbers. One has:2N1009N12/Nπ /N• {1,2,3}and{1,3}are both finite proper subsets ofN.• {1,3} ⊂ {1,2,3}Nis an infinite set.12
Definition 17.Aset of outcomesis the set of all possibilities that could result from some activity orexperience.Example 31.For example, if we toss a coin and observe which side faces up, there are two possibleoutcomes, heads (H) or tails (T). The set of outcomes of tossing a coin once can be writtenS={H, T}.Example 32.Another example is rolling a die with usual faces 1 through 6 and observing which face isup when the die lands. The set of outcomes isS={1,2,3,4,5,6}.Example 33.(Two Dice: Indistiguishable vs. Distinguishable) See Example 1 in Section 6.1 of [1].When we don’t want to list the individual elements of a set, but instead have some other way ofdescribing those elements, we can useset-builder notation. For example, the setB={0,2,4,6,8}might be alternatively written (in set-builder notation) asB={n|nis a nonnegative even integer less than 10}.This formal statement is read as:Bis the set of allnsuch thatnis a nonnegative even integer less than 10.”Note that the vertical line “|” means “such that” in this setting. A third way of writing this same set is:B={n|0n <10, nan even integer}.AVenn diagramis a representation of sets as regions which allows us to visualize sets and relations

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