20130919010904sma6014chap1.7student(binary op)

8/11/2019 20130919010904SMA6014Chap1.7student(binary op)

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1.7 Binary Operations

A binary operation(or just operation) on a setAis a rule which assigns to each ordered pair(a,b) of elements

of A exactly one element a b in A.

Example

1. The usual addition(+) on Z, R, C, R+, Z+.

2. The usual multiplication (*) on Z, R, C, R+, Z+.

AAA X:That is:


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(2) a b must be uniquely defined.

Three aspects of the definition that need to be stressed:

(1) a b is defined for every ordered pair (a,b) of

elements ofA.Addition (+) onM(R) is not defined.

M(R) -- the set of all matrices with real entries.

A+B is not defined for an ordered pair(A,B) of matrices

having different numbers of rows or of columns.

Suppose we define an operationon Rsuch that

,,anyforRba

.issquaresenumber whotheisabba

defineduniquelynotis since

82 is 4 or -4

Hence addition (+) onM(R) is not an operation

.onoperationannotisHence R


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(3) If a and b are inA, then a b must also be inA.

-closed under operation.

Suppose we have a setA={0,1,2,3,4}

+ onAis not an operation sinceA is not closed under +

ex: 2+4 =6 A

Example1. Is addition(+) an operation on R*? R*-Nonzero real numbers.

Solution

Hence addition (+) onA is not an operation.


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Commutative

An operationon a set A iscommutat ive if (and only if)

a b = b a .,allfor Aba

Example

Is the operation below commutative?

1.

be an operation onZ+

such that ,,for

Zba

abequals the smaller of aand b or the commonvalue ifa=b.

Solution

Remark

If the question is: Is a commutative operation .?

Need to check whether it is an operation first!!

Properties of Operation


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be an operation onZ+such that2.

a b = a

Solution

Associative OperationAn operationon a setA is associative if (and only if)

(a b) C= a (b C )

Example

Addition on R is associative, but division is not.

Ex. (3/4)/5 = 3/20 3/(4/5) =15/4


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Identitiy Element

Let be an operation on a setA.

If there is an elemente in A with the property that

e a = a and a e =a for every element a inAthene is called an identity or neutral element with

respect to the operation

Example

0 is the identity element for addition in R.

1 is the identity element for multiplication in R.

Remark

An identity element is unique.

That is, it is the same for all element of a set.


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Inverse Element

Letbe an operation on a set A.

If a is an element inA, andx is an element ofA with theproperty that

a x = e and x a =e

thenx is called an inverse of a.

Example

-ais the inverse of afor addition in R.

1/ais the inverse of afor multiplication in R )0( a

RemarkAn inverse element is not unique in a set but it is unique

for each element.

The inverse of a is denoted by a-1.

20130919010904sma6014chap1.7student(binary op)

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