Question

Let A = Q x Q , Q being the set of rationals . Let ‘*’ be a binary operation
on A , defined by (a, b) * (c , d) = ( ac , ad + b) . Show that
(i) ‘*’ is not commutative (ii) ‘*’ is associative
(iii The Identity element w.r.t ‘*’ is ( 1 , 0)

Answer 1

help me plzz

Answer 2

(i) (a,b) * (c,d) = (ac, ad + b)
(c,d) * (a,b) = (ca,cb + d)
(ac, ad + b) \(\neq\) (ca, cb + d)
therefore it is not commutative

Answer 3

thanks bro and how i solve (ii) part?

Answer 4

compute (a,b) * ((c,d) * (e,f))
compute ((a,b) * (c,d)) * (e,f)
compare the two ordered pairs if they are equal

Answer 5

(a,b) * ((c,d) * (e,f)) = (a,b) * (ce, cf + d) = (ace, a(cf + d) + b)
((a,b) * (c,d)) * (e,f) = (ac, ad + b) * (e,f) = (ace, acf + (ad + b))
the two ordered pairs are the same, therefore * is associative

Answer 6

thnanks again @sirm

Answer 7

yw