question_answer 54)

Let A = N × 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 element for * on A, if any.  

Answer:

Let (a, b) * (c, d) = (a + c, b + d)       = (c + a, d + b)       = (c, d) * (a, b)        is commutative.       Also ((a, b) * (c, d) * (e, f)             = (a + c, b + d) * (e, f)             = ((a + c) + e, (b + d) + f)             = (a + (c + e), b + (d + f))             = (a, b) * (c + e, d + f)             = (a, b) * ((c, d) * (e, f))              is associative.       To find the identity element.       Let (x, y) be the identity element.       Let (x, y) be the identity element.                          a + x = a and b + y = 6        x = 0 and y = 0        is the identity element.       But (0, 0)  so Identity element does not exist.