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#### jakncoke

##### Active member

- Jan 11, 2013

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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 .

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- Thread starter
- #1

- Jan 11, 2013

- 68

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 .

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- #2

- Jan 11, 2013

- 68

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.