Question

Consider the binary operation * defined on the set A = {a, b, c, d} by the following table:

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

Is it commutative and associative?

Answer 1

From the table

b * c = b

c * b = d

So, the binary operation is not commutative.

To check whether the given operation is associative.

Let a, b, c ∈ A.

To prove the associative property we have to prove that a * (b * c) = (a * b) * c

From the table,

L.H.S: b * c = b

So, a * (b * c) = a * b = c  ........(1)

R.H.S: a * b = c

So, (a * b) * c = c * c = a  ........(2)

(1) ≠ (2).

So, a * (b * c) ≠ (a * b) * c

∴ The binary operation is not associative.