Theorem 1 (Left & Right Cancellation Laws)
If G is a group with binary operation *, then the left and right cancellation laws hold in G, that is
a*b = a*c implies b = c, and
b*a = c*a implies b = c for all a,b,c in G
Similarly, we can prove the right cancellation law.
b*a = c*a implies b = c for all a,b,c in G
Proof:
(To prove the left cancellation law)
To show if a*b = a*c then b = c
Let a*b = a*c
Since G is a group, G3 is satisfied. Then, for all a' in G such that,
(To prove the left cancellation law)
To show if a*b = a*c then b = c
Let a*b = a*c
Since G is a group, G3 is satisfied. Then, for all a' in G such that,
a'*(a*b) = a'*(a*c) --G3
(a'*a)*b = (a'*a)*c --G1,* associative
e*b = e*c --by def. of a' in G3
b = c
(a'*a)*b = (a'*a)*c --G1,* associative
e*b = e*c --by def. of a' in G3
b = c
Similarly, we can prove the right cancellation law.
