Question

Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d).

Show that * is commutative and associative. Find the identity for * oh A, if any.

Answer 1

A = N x N
Binary operation * is defined as
(a, b) * (c, d) = (a + c, b + d).
(a) Now, (c, d) * (a, b) = (c + a, d + b)
= (a + c, b + d)
⇒ (a, b) * (c, d) = (c, d) * (a, b)
∴ This operation * is commutative.

(b) Next
(a, b) * [(c, d) * (e, f)] = (a, b) * (c + e, d + f)
= [(a + c + e), (b + d + f)]
and
[(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f)
= [(a + c + e), (b + d + f)]
⇒ (a, b) * [(c, d) * (e, f) = [(a, b) * (c, d)] * (e, f)
∴ The given binary operation is associative.

(c) Identity element does not exist, because there is no e and e’ in N x N such that (a, b) * (e, e’) = (a + e, b + e’) = (a, b).