3 Binary Operations - Arkansas Tech University

1 Arkansas Tech UniversityMATH 4033: Elementary Modern AlgebraDr. Marcel B. Finan3 Binary OperationsWe are used to addition and multiplication of real numbers. These operationscombine two real numbers to generate a unique single real number. So wecan look at these Operations as functions on the setR R={(a, b) :a Rand b R}defined by+ :R R R(a, b) a+band :R R R(a, b) a bThese Operations are examples of a Binary operation. The general definitionof a Binary operation is as operationon a set S is a mapping that assigns to each orderedpair of elements ofSa uniquely determined element is, :S S Sis a mapping. The setSis said to beclosedunder the operation .The image (a, b) will be denoted bya and multiplication are Binary Operations on the setZof integersso that this set is closed under these Operations . However,Zis not closedunder the operation of division since 1 2 is not an ordered pair statement in Definition is critical.

2 For example,consider the Binary operation defined on the setNbya b= 3 = 23= 8 and 3 2 = 32= is, 2 36= 3 (Cayley s Tables)The idea of a Binary operation is just a way to produce an element of a setfrom a given pair of ordered elements of the same set. In the case of a finiteset we could list the rule in a table which we ll call amultiplication tableorCayley s table. For example, the following is the multiplication table of abinary operation :{a, b} {a, b} {a, b}.*abaabbbaIn studying Binary Operations on sets, we tend to be interested in thoseoperations that have certain properties which we discuss Binary operation on a setSis said to beassociativeif it satisfies theassociative law:a (b c) = (a b) cfor alla, b, c associative property allows us to speak ofa b cwithout having toworry about whether we should find the answer toa bfirst and then thatanswer multiplied by c rather than evaluateb cfirst and then multiply a with that answer.

3 Which ever way we process the expression we end upwith the same element of the set. Note though that it does not say we cando the product in any order ( bandb amay not have the same value).Example The Operations + and onRare The operation onRis not associative since 2 (3 4)6= (2 3) 4.(Notice that if the associative law fails for just one triple (a, b, c) then theoperation is not associative).3. The operation defined bya b=abon the setNis not associative since2 (3 2) = 512 and (2 3) 2 = Binary operation on a setSis said to becommutativeif it satisfies thecondition:a b=b a2for alla, b, this case, the order in which elements are combined doesnot a set with a Binary operation is given by a Cayley s table then theoperation is commutative if and only if equal elements appear in all positionsthat are symmetrically placed relative to the diagonal from upper left to lowerright.

4 That is, to check whether an operation defined by a Cayley s table iscommutative, simply draw a diagonal line from upper left to lower right, andsee if the table is symmetric about this line. For example, the operation defined by the table below is commutative.*abcdaabcdbbcdaccdabddabcExa mple Binary Operations of addition and multiplication onRare both commu-tative. However, the Binary operation of subtraction onRdoes not satisfythe commutative law since 5 76= 7 Binary operation onRdefined bya b=a+b 1 is commutative sincea b=a+b 1 =b+a 1 =b that the Binary operation onRdefined bya b= 1+abis commutativebut not any real numbersaandbwe havea b= 1 +ab= 1 +ba=b awherewe used the fact that multiplication inRis commutative. Now, by lettinga= 0, b= 1,andc= 1 thena (b c) =a 0 = 1 and (a b) c= 1 c= , is not a set on which there is a Binary operation . An elementeof thisset is called aleft identityif for alla S,we havee a= , anelementeis aright identityifa e=afor eacha a Binary operation on a There might be left identities which are not right identities and vice-versa.

5 For example, the operationa b=aon the setRhas 2 as a rightidentity which is not a left identity. The setRwith the operationa b=bhas 2 as a left identity which is not a right There might be many left or right identity elements. The setRwiththe operationa b=a,every number is a right identity. With the operationa b=b, every number is a left There might be no left or right identity elements. For example, theset{2,3,4, }has no left or right identity elements under the operationa b=a bWe tend to be familiar with the situation in which there is a unique iden-tity. As soon as an operation has both a left and a right identity, they arenecessarily unique and equal as shown in the next a set with a Binary operation that has a left identity elemente1anda right identity elemente2thene1=e2= Sbe a left identity element ande2 Sbe a right identity e2(since e2is a right identity)=e2(since e1is a lef t identity)Definition element which is both a right and left identity is called theidentityelement(Some authors use the term two sided identity.)

6 Thus, an elementis an identity if it leaves every element that an identity (left or right or both) for one operation does not haveto be an identity for another operation. Think of addition and multiplicationon the reals where the identities are 0 and 1 operationa b=a+b 1 on the set of integers has 1 as an identityelement since 1 a= 1 +a 1 =aanda 1 =a+ 1 1 =afor all that the operationa b= 1+abon the set of integersZhas no an identity element then we must havea e=afor alla par-ticular, 1 e= this imply that 1 +e= 1 ore= 2 0 = 16= 2thenedoes not a set has an identity element with respect to a Binary operationon the set, it is then in order to raise the question of that an operation on a setShas an identity there is an elementb Ssuch thata b=ethenbis called arightinverseofa. Similarly, ifb a=ethenbis called aleft An element can have no left or right inverses.

7 For example, the number 2has no left or right inverse with respect to multiplication on the set of There might be a left inverse which is not a right inverse and vice example, consider the setM(Z) of all functions from the set of integersinto itself. Then the operation of composition is a Binary operation onM(Z).Consider the two functionsf(n) = 2nandg(n) ={n2ifnis even4 ifnis odd5 Then (g f)(n) =nfor alln Z. That is,gis a left inverse ,since(f g)(n) ={nifnis even8 ifnis oddthengis not a right inverse sincef g6= ZSuppose that an elementa Shas both a left inverse and a right inversewith respect to a Binary operation onS. Under what condition are the twoinverses equal?Theorem a set with an associative Binary operation and identity , b, c Sbe such thata b=eandc a= ,b=e b= (c a) b=c (a b)=c e=cDefinition both a left and right inverse then we say thatahastwo-sidedinverseor simply the operation on the set of integers defined bya b=a+b will show that each integer has an inverse under this operation.}}

8 Indeed,letxbe an integer. Letybe a right inverse ofx. Thenx y= is,x+y 1 = forywe findy= x+ is also a left inverse ofxsince ( x+ 2) x= x+ 2 +x 1 =