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Solution
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Then for all a,b,c∈GLet G be a group.
(i) a∗b=a∗c⇒b=c (Left cancellation law)
(ii) b∗a=c∗a⇒b=c (Right cancellation law)
Proof:
a∗b=a∗c
Pre multiplying by a−1, we get
a−1∗(a∗b)=a−1∗(a∗c)
⇒(a−1∗a)∗b=(a−1∗a)∗c⇒e∗b=e∗c
(i.e)b=c
(ii) b∗a=c∗a
Post multiplying by a−1, we get
(b∗a)∗a−1=(c∗a)∗a−1
b∗(a∗a−1)=c∗(a∗a−1)
(i.e.,) b∗e=c∗e⇒b=c
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