Question

Consider the binary operation '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

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 that o is commutative on S.
Associativity:

\[a o \left( b o c \right) = a o c\]
               \[ = a\]
\[\left( a o b \right) o c = a o c\]
                \[ = a\]
\[\text{Thus},\]
\[a o \left( b o c \right) = \left( a o b \right) o c \forall a, b, c \in S\]

So, 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.

\[\Rightarrow x o b = b o x\]
\[ = x, \forall x \in S\]

So, b is the identity element.

Finding inverse elements :-

\[\text{In the first row, we don't haveb, i.e. there does not exist an elementxsuch thata} o x = x o a = b . \]
\[So, a^{- 1} \text{does not exist}.\]
\[b o b = b\]
\[ \Rightarrow b^{- 1} = b\]
\[c o d = b\]
\[ \Rightarrow c^{- 1} = d\]
\[d o c = b\]
\[ \Rightarrow d^{- 1} = c\]