Consider the binary operation ^* and o defined by the following t… - 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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