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Let $(A, *)$ be a semigroup, Furthermore, for every $a$ and $b$ in $A$, if $a \neq b$, then $a*b \neq b*a$.

1. Show that for every $a$ in $A$, $a*a=a$
2. Show that for every $a$, $b$ in $A$, $a*b*a=a$
3. Show that for every $a,b,c$ in $A$, $a*b*c=a*c$
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a. Let $a * a = b$. $(a * a) * a = b*a$. Since $(A, *)$ is a semigroup, $*$ is closed and associative. So, $(a *a) * a = a * (a*a) \implies a * b = b * a$, which is possible only if $a = b$. Thus we proved $a * a = a.$

b. Let $(a * b)*a = c \\ \implies (a * b) * a * a = c * a \\ \implies a * b * a = c * a \\ \implies c * a = a.$

Similarly, $a * (a * b * a ) = a * c \\ \implies a * a * (b * a) = a * c \\ \implies a * (b * a) = a * c \\ \implies a * c = a = c * a.$

So, $c = a$.

c.  Let $(a * b)*c = d. \\ \implies (a * b) * c * c = d * c \\ \implies a * b * c = d * c \\ \implies d * c = d.$

Similarly, $a * (a * b * c ) = a * d \\ \implies a * a * (b * c) = a * d \\ \implies a * (b * c) = a * d \\ \implies a * d = d.$

Thus $d * c = a * d = d$

Now $c * d *c = c * a * d = c * d \\\implies c = c * a * d = c * d$

and

$d * c * a = a * d * a = d * a \\\implies d * c * a = a = d * a$

So,

$a * c = (d*a)*(c*d) \\= d*(a*c)* d = d.$

Thus, $a*b*c = a * c.$

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Suppose this type of question comes in exam

how to solve means how to start initially ???..

I am finding it difficult...:(
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Watch nptel video lectures on mathematics by prof Kamala krithivasan.If you are weak only in group theory just watch lectures 35,36 and 37.

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## a∗b∗a=c∗a⟹c∗a=a????????????????

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I think for b) the proof should be as follows:

$Let, a*b*a=c\\ \bullet (a*b*a)*a = c*a\\\implies (a*b)*a*a =c*a\\\implies (a*b)*a =c*a\\\implies c=c*a ----\ (1)\\\bullet a*(a*b*a) = a*c\\\implies (a*b*a)=a*c\\\implies c=a*c----\ (2)\\ \text{from (1) and (2): }a*c=c*a\\\therefore c=a \implies a*b*c = a$
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@krish

yes right :)

in solution c   given in best answer

Thus d∗c = a∗d  =d

Now c∗d∗c = c∗a∗d = c∗d

⟹c= c∗a∗d= c∗d ( here how the value c*d*c in previous step become c ? )

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My proof would be:
Consider $a*b*c = d$
$d*(a*c) = a*b*(c*a*c) = a*b*c$
$(a*c)*d=a*c*(a*b*c)=(a*c*a)*b*a=a*b*c$
So, $d*(a*c)=(a*c)*d\implies d=a*c=a*b*c$