Consider the binary operation * and o defined by the following ta… - Vidyadip

Question

Consider the binary operation * and o defined by the following tables on set S = {a, b, c, d}.

*	a	b	c	d
a	a	b	c	d
b	b	a	d	c
c	c	d	a	b
d	d	c	b	a

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Answer

Commutativity:The table is symmetrical about the leading element. It means * is commutative on S.
Associativity:
$a * (b * c) = a * d$
$= d$
$(a * b) * c = b * c$
$=d$
Therefore,
a * (b * c) = (a * b) * c $\forall\text{ a, b, c}\in\text{S}$
So, * is Associative on S.
Finding identity element:
We observe that the first row of the composition table coincides with the top-most row and the first column coincides with the left-most column.
These two intersect at a.
$\Rightarrow x * a = a * x = x,$ $\forall\text{ x}\in\text{S}$
So, a is the identity element:
$a * a = a$
$\Rightarrow a^{-1} = a$
$b * b = a$
$\Rightarrow b^{-1} = b$
$c * c = a$
$\Rightarrow c^{-1} = c$
$d * d = a$
$\Rightarrow d^{-1} = d$

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