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Properties and Relations

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Closure Property for AdditionFor any two numbers a and b, a+b is unique and a+b is contained in the set of numbers.For what sets of numbers is this property valid?Find a set of numbers for which this property is not valid.

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Closure Property for MultiplicationFor any two numbers a and b, ab is unique and ab is contained in the set of numbers.For what sets of numbers is this property valid?Find a set of numbers for which this property is not valid.

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Commutative Property for AdditionFor any two numbers a and b, a+b = b+a.For what sets of numbers is this property valid?

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Commutative Property for MultiplicationFor any two numbers a and b, ab = ba.For what sets of numbers is this property valid?

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Associative Property for AdditionFor any three numbers a, b, and c, a + (b + c) = (a + b) + c.For what sets of numbers is this property valid?

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Associative Property for MultiplicationFor any three numbers a, b, and c, a(bc) = (ab)c.For what sets of numbers is this property valid?

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Distributive PropertyFor any three numbers a, b, and c, a(b + c) = ab + ac.For what sets of numbers is this property valid?

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Additive Identity PropertyFor any number a, a + 0 = 0 + a = a. Zero is called the additive identity. For what sets of numbers is this property valid?

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Additive Inverse PropertyFor any number a, there exists a number a (read the opposite of a) such that a + a = 0. For what sets of numbers is this property valid?

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Multiplicative Identity PropertyFor any number a, a(1) = 1(a) = a. One is called the multiplicative identity. For what sets of numbers is this property valid?

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Multiplicative Inverse PropertyFor any number a, there exists a number 1/a (read the reciprocal of a) such that a(1/a) = 1. For what sets of numbers is this property valid?

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Reflexive PropertyFor any number a, a = a. For what sets of numbers is this property valid?

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Symmetric PropertyFor any two numbers a and b, if a = b then b = a. For what sets of numbers is this property valid?

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Transitive PropertyFor any three numbers a, b, and c, if a = b and b = c, then a = c. For what sets of numbers is this property valid?

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Identity RelationBecause equality exhibits the reflexive, symmetric, and transitive properties, it is called an equivalence relation.

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Equivalence RelationAny relation that has the reflexive, symmetric, and transitive properties is an equivalence relation.

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Lets Explore RelationsDetermine if the relation has the same color of eyes as on the set of people who live in the USA is an equivalence relation.Reflexive?YesSymmetric?YesTransitive?Yes18

Lets Explore RelationsDetermine if the relation has the same color of eyes as on the set of people who live in the USA is an equivalence relation.Reflexive?YesSymmetric?YesTransitive?Yes19Therefore, this IS an equivalence relation.

Lets Explore RelationsDetermine if the relation is a classmate of on the set of students currently enrolled in MATH 517 is an equivalence relation.Reflexive?NoSymmetric?YesTransitive?Yes20

Lets Explore RelationsDetermine if the relation is a classmate of on the set of students currently enrolled in MATH 517 is an equivalence relation.Reflexive?NoSymmetric?YesTransitive?Yes21Therefore, this is NOT an equivalence relation.

Lets Explore RelationsDetermine if the relation is similar to on the set of quadrilaterals is an equivalence relation.Reflexive?YesSymmetric?YesTransitive?Yes22

Lets Explore RelationsDetermine if the relation is similar to on the set of quadrilaterals is an equivalence relation.Reflexive?YesSymmetric?YesTransitive?Yes23Therefore, this IS an equivalence relation.

Lets Explore RelationsDetermine if the relation has the same or more volume than on the set of 3-D figures is an equivalence relation.Reflexive?YesSymmetric?NoTransitive?Yes24

Lets Explore RelationsDetermine if the relation has the same or more volume than on the set of 3-D figures is an equivalence relation.Reflexive?YesSymmetric?NoTransitive?Yes25Therefore, this is NOT an equivalence relation.

Lets Explore RelationsDetermine if the relation has the same number of factors as on the set of prime numbers is an equivalence relation.Reflexive?YesSymmetric?YesTransitive?Yes26

Lets Explore RelationsDetermine if the relation has the same number of factors as on the set of prime numbers is an equivalence relation.Reflexive?YesSymmetric?YesTransitive?Yes27Therefore, this IS an equivalence relation.

Lets Explore RelationsDetermine if the relation have a common factor greater than 1 on the set {2, 3, 4, } is an equivalence relation.Reflexive?YesSymmetric?YesTransitive?No28

Lets Explore RelationsDetermine if the relation have a common factor greater than 1 on the set of composite numbers is an equivalence relation.Reflexive?YesSymmetric?YesTransitive?No29Therefore, this is NOT an equivalence relation.

Lets Explore Relations

SetRelationReflexive?Symmetric?Transitive?Equivalence Relation?PeopleIs a sister ofPeopleIs a sibling ofTrianglesAre congruentLines in a planeHave at least one point in commonIntegersDifference is divisible by 2Natural numbersDivisible by

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Lets Explore Relations

SetRelationReflexive?Symmetric?Transitive?Equivalence Relation?PeopleIs a sister ofNoNoYesNoPeopleIs a sibling ofNoYesYesNoTrianglesAre congruentYesYesYesYesLines in a planeHave at least one point in commonYesYesNoNoIntegersDifference is divisible by 2YesYesYesYesNatural numbersDivisible byYesNoYesNo

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Now Its Your TurnFind some sets and relations that represent equivalence relations.32