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 $S = {a, b, c, d}.$

$o$	$a$	$b$	$c$	$d$
$a$	$a$	$a$	$a$	$a$
$b$	$a$	$b$	$c$	$d$
$c$	$a$	$c$	$d$	$b$
$d$	$a$	$d$	$b$	$c$

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Answer

Commutativity:The table is symmetrical about the leading element. It means that $o$ is commutative on $S.$
$\text{a o (b o c) = a o c}$
$= a$
$\text{(a o b) o c = a o c}$
$= a$
thus,
$\text{a o (b o c) = (a o b)} o \text{c }\forall\text{ a, b, c}\in\text{S}$
Associativity:
Therefore, $o$ is associative on $S.$
Finding identity element:
We observe that the second 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 $b.$
Implies that $\text{x o b = b o x}$
$=\text{x, }\forall\text{ x}\in\text{S}$
Therefore,
$b$ is the identity element.
Finding inverse elements:
In the first row, we don't have $b,$
i.e. there does not exist an element $x$ such that $\text{a o x = x o a = b}.$
Therefore,
$a^{-1}$ does not exists.
$\text{b o b = b}$
Implies that $b^{-1 }= b$
$\text{c o d = b}$
Implies that $c^{-1} = d$
$\text{d o c = b}$
Implies that $d^{-1} = c$

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