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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let $(X,*)$ be a semigroup. Furthermore, for every $a$ and $b$ in $X$, if $a\ne b$, then $a*b\ne b*a$.
Based on the defined semigroup, choose the correct equalities from the options given below.
A. For every $a$ in $X,\ a*a=a$
B. For every $a,b$ in $X,\ a*b*a=a$
C. For every $a,b,c$ in $X,\ a*b*c=a*c$
Based on the defined semigroup, choose the correct equalities from the options given below.
A. For every $a$ in $X,\ a*a=a$
B. For every $a,b$ in $X,\ a*b*a=a$
C. For every $a,b,c$ in $X,\ a*b*c=a*c$
Solution
Given condition means if two elements commute, then they must be equal.For A:
$a*(a*a)=(a*a)*a$ by associativity.
So, $a$ and $a*a$ commute.
Hence, $a=a*a$.
So, A is true.
For B:
$a*(a*b*a)=(a*a)*b*a=a*b*a$
$(a*b*a)*a=a*b*(a*a)=a*b*a$
So, $a$ and $a*b*a$ commute.
Hence, $a*b*a=a$.
So, B is true.
For C:
Using B, $c*a*c=c$ and $a*c*a=a$.
Now,
$(a*b*c)(a*c)=a*b*(c*a*c)=a*b*c$
$(a*c)(a*b*c)=(a*c*a)*b*c=a*b*c$
So, $a*b*c$ and $a*c$ commute.
Hence, $a*b*c=a*c$.
So, C is also true.
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