BINARY OPERATIONS 3

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Some important words:

Not everywhere define - Operation * is called not everywhere defined on S it - no element can be assigned to each possible ordered pairs.

Not well defined - Operation * is called not well defined on S it - several element of S are assigned to S (ambiguity).



Definition 5 ( Binary Algebraic Structure)

A binary algebraic structure, denoted by (S,*) is a set S together with a binary operation * on S.

Example : (Z,*) , (Q,*)


Definition 6 (Identity element for *)

Let (S,*) be a binary structure. An element e of S is an identity element for * if

e*s = s*e = s.

Example 1 :

• 0 is identity for + (addition) on R. Since
  

0+a = a+0 = a for all a in R.

• 1 is identity for . (multiplication) on R. Since

1.a = a.1 = a for all a in R.

Example 2 :

Determine the identity of (Q,*) where * is defined by a*b = ab/2.

Solution :

Let a, b element of Q

LHS :

a*b = a

ab/2 = a

ab = 2a
b = 2 element of Q

RHS :
b*a = a
ba/2 = a
ba = 2a
b = 2 element of Q

Since LHS = RHS and 2 is element of Q, then b = 2 is identity of (Q,*).


Example 3 :

Determine whether (Z,*), where * is defined by a*b = a-b+1, has an identity.

Solution :

LHS :
a*b = a
a-b+1 = a
-b+1 = 0
b = 1 element of Z

RHS :
b*a = a
b-a+1 = a
b+1 = 2a
b = 2a-1 element of Z

Since LHS and RHS is not equivalent, therefore, there is no identity for (Z,*) .