Question

Consider the binary operation * 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

Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.

Answer 1

Commutativity:
The table is symmetrical about the leading element. It means * is commutative on S.

Associativity:

\[a * \left( b * c \right) = a * d\]
                  \[ = d\]
\[\left( a * b \right) * c = b * c\]
                  \[ = d\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c \forall a, b, c \in S\]

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.

\[\Rightarrow x * a = a * x = x, \forall x \in S\]

So, a is the identity element.

Finding inverse elements :- 

\[a * a = a\]
\[ \Rightarrow a^{- 1} = a\]
\[b * b = a\]
\[ \Rightarrow b^{- 1} = b\]
\[c * c = a\]
\[ \Rightarrow c^{- 1} = c\]
\[d * d = a\]
\[ \Rightarrow d^{- 1} = d\]