Algebra I – Terminology Sets and Expressions - Quick Guides | Mathematics

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Study GuideAlgebra I–Terminology Sets and Expressions1.Set TheoryWhat Is a Set?Asetis simply acollection of objects or numbersgrouped together.For example:{1, 2, 3}This set contains the numbers1, 2, and 3.When we say3 is part of the set, we write it using the symbol∈, which means“is an element of”or“belongs to.”Example:3∈{1,2,3}This reads as:“3 is an element of the set {1,2,3}.”Special Types of SetsSubsetsAsubsetis a set that is completely contained inside another set.Example:{1,2} is a subset of {1,2,3}The larger set contains all the elements of the smaller set.There are two types of subsets:Proper Subset

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Study GuideAproper subsetis a subset that doesnot contain all elementsof the original set.Example:{1,2}⊂{1,2,3}Here the symbol⊂means“proper subset of.”Improper SubsetIf a subset isexactly the same as the original set, it is called animproper subset.Example:{1,2,3}⊆{1,2,3}The symbol⊆means“subset of (possibly equal).”Universal SetTheuniversal setis thecomplete set of all elements being consideredin a particular situation.Think of it as thebig group that contains everything we are studying.Empty Set (Null Set)Theempty set(also called thenull set) is a set thatcontains no elements.It is written as:∅or { }Important note:The empty set isnot written as {∅}, because that would mean a set containing one element (theempty set itself).Important FactBoth theuniversal setand theempty setaresubsets of every set.Describing SetsThere aretwo main ways to describe a set.

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Study Guide1. Rule MethodIn this method, wedescribe the elements of a set using a rule or condition.Example:{x : x > 3, x is a whole number}This means:Allwhole numbers greater than 3So the set would be:{4, 5, 6, ...}Since the numbers continue forever, this type of set is often represented using anumber lineratherthan listing all values.2. Roster MethodIn theroster method, we simplylist all the elements in the set.Example:{1,2,3}This set contains the numbers1, 2, and 3 only.The same set can also be written using a rule.Example:{x : x < 4, x is a natural number}or{x : 0 < x < 4, x is a whole number}Both descriptions represent the set:

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Study Guide{1,2,3}Incorrect Example{x : 0 < x < 4}This rule includesall numbers between 0 and 4, such as:1, 1.5, 2, 2.7, 3.2, etc.So it doesnot correctly describe the set {1,2,3}.Types of SetsFinite SetsAfinite sethas alimited (countable) number of elements.Example:{a, b, c, d, e}This set containsfive elements, so it is afinite set.Infinite SetsAninfinite sethasunlimited elements.Example:{1,2,3,...}The numbers continue forever, so this is aninfinite set.Comparing SetsEqual Sets

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Study GuideTwo sets areequalif they containexactly the same elements, even if the order is different.Example:{1,2,3} = {3,2,1}The elements are the same, so the sets are equal.Equivalent SetsTwo sets areequivalentif they have thesame number of elements, even if the elements aredifferent.Example:{1,2,3} and {a,b,c}Each set containsthree elements, so they are equivalent.Venn DiagramsAVenn diagram(also calledEuler circles) is a visual way to show relationships between sets.Figure 1.A Venn diagramExplanation of the diagram:

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Study Guide•Arepresents the elements inside theleft circle.•Brepresents the elements inside theright circle.•Crepresents theoverlapping area, which contains elements that belong toboth A and B.Operations with SetsUnion of SetsTheunionof two sets combinesall elements from both sets.If an element appears in both sets, it is writtenonly once.The symbol used for union is:∪Example:IfA = {1,2,3}B = {3,4,5}ThenA∪B = {1,2,3,4,5}Notice that3 appears only once.Theunion of two setscombinesall the elements from both setsinto one set.The symbol used for union is:∪When forming the union,duplicate elements are written only once.

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Study GuideExample 1Find the union:{1,2,3}∪{3,4,5}Step 1:Write all elements from both sets.Set A→{1,2,3}Set B→{3,4,5}Step 2:Combine them and remove duplicates.{1,2,3}∪{3,4,5} ={1,2,3,4,5}Explanation:The number3 appears in both sets, but it is writtenonly oncein the union.So, the union containsall the elements from both sets.Intersection of SetsTheintersection of two setscontainsonly the elements that appear in both sets at the sametime.The symbol used for intersection is:∩Example 2Find the intersection:{1,2,3}∩{3,4,5}Step 1:Look for elements that appear inboth sets.Set A→{1,2,3}

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Study GuideSet B→{3,4,5}Step 2:Identify the common element.The number3 appears in both sets.So,{1,2,3}∩{3,4,5} ={3}Explanation:The intersection containsonly the shared element, which is3.Using Venn Diagrams to Understand SetsTo better understand how sets relate to each other, we can use aVenn diagram.Figure 2.Intersection of set A and set BUnderstanding the DiagramIn the diagram:•Set Acontains the numbers1 and 2.

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Study Guide•Set Bcontains the numbers4 and 5.•Theoverlapping regioncontains3, which belongs toboth sets.From this diagram:Union of A and B:A∪B = {1,2,3,4,5}Intersection of A and B:A∩B = {3}Theunion includes every number shown in the diagram, while theintersection includes onlythe overlapping element.Example 3Find the intersection:{1,2,3}∩{4,5}Step 1:Check for elements that appear in both sets.Set A→{1,2,3}Set B→{4,5}Step 2:Compare the elements.There areno common elementsin these two sets.So,{1,2,3}∩{4,5} =∅What Does∅Mean?The symbol∅represents theempty set(also called thenull set).

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Study GuideIt means thatthe sets do not share any elements.So, when two sets haveno common members, their intersection is theempty set.2. Quiz: Set Theory1. QuestionWhich of the following symbols represents“is an element of”?Answer Choices•∈•⊂•⊆Correct Answer:∈Why This Is CorrectThe symbol∈means“is an element of”in set theory. It shows that a particular object belongs to aset.Example:3∈{1, 2, 3, 4}This means3 is an element of the set {1,2,3,4}.The other symbols mean different things:•⊂meansproper subset•⊆meanssubset (possibly equal)

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