A binary operation

is defined on the set

by

for all

. Determine which of the properties (Associative, Identity, Inverse, Commutative) are satisfied by

Solution:

Let

, and

be any three elements of

(distinct of not). Then

, while

. Thus (S,*) is associative. Now (S,*) has no identity since for every element

, it follows that

and so it is impossible for

and

. Since (S,*) has no identity, the question of inverses does not apply here. Certainly,

is not commutative since

while

.

Question:

I suppose

is an identity element. Since (S,*) has no identity why it contradict itself by saying "(S,*) has no identity since for every element "

"?