Question

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as

a * b = \(\left\{\begin{array}{cl} a+b, & \text { if } a+b<6 \\ a+b-6, & \text { if } a+b \geq 6 \end{array}\right.\)

Show that 0 is the identity for this operation and each element of the set is invertible with 6 – a being the inverse of a.

Answer 1

(i) e is the identity element, if a * e = e * a = a.
a * 0 = a + 0, 0 * a = 0 + a = a.
⇒ a * 0 = 0 * a = a.
∴ 0 is the identity for the operation.

(ii) b is the inverse of a if a * b = b * a = e.
Now, a * (6 – a) = a + (6 – a) – 6 = 0
and (6 – a) * a = (6 – a) + a – 6 = 0.
Hence, each element of a of the set is invertible with inverse 6 – a.