Let S be a set be an associative binary operation on S wan

Let S be a set & be an associative binary operation on S w/an identity element e. Show that any element has at most 1 inverse.

Solution

Let S be a set and * be an associate binary operation on S and e be an identity element in S

Then for any a , b , c S , we have a * (b * c) = (a * b) * c

and a*e = e*a = a

Now we show that every element of S has atmost one inverse.

Assume that a₁ and a₂ are the inverses of a S

then we have a * a₁ = a₁*a = e ------------ ( 1 ) and   a*a₂ = a₂*a = e -------------( 2 )

From ( 1 ) and ( 2 )

e = a*a₁ = a*a₂

   => a*a₁ = a*a₂   multiply by a^-1 on both sides

   => a^-1 * ( a*a₁ )= a^-1 * ( a*a₂ )

   => ( a^-1 * a* )a₁ = (a^-1 * a )*a₂    from associative law

   => (e)*a₁ = (e)*a₂

   => a₁ = a₂

Hence the inverse any a S is unique