A binary operation is defined on the set by for all . Determine which of the properties (Associative, Identity, Inverse, Commutative) are satisfied by
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 .
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 " "?
and make sense, but what is ?