Putnam 2012: B2

[jakncoke]

Let * be a commutative, assosiative binary operation on a set S with the property that
for all x,y $\in S$, there exists a z $\in S$, such that x*z = y. Prove that if a*c = b*c then a = c .

 

[jakncoke]

Spoiler

Assume a*c = b*c and a is not equal to b.
then for b,a there exists an assosiated non identity element p such that b = p*a. If there exists an identity element such that b = p*a, then b = a and we derive a contradiction.

So assume so p*a $\not = a$ (p is non identity)
then b*c = (p*a)*c = a*c
by assosiativity, b*c = p*(a*c)
by commutativity, b*c = (a*c)*p (i subsituted a*c = b*c here)
so b*c = (b*c)*p (i subsituted a*c = b*c here)
which is a contradiction if p is a non identity.