1. ## number

Can someone pls guide me through . Thanks

Q : The binary operation , * , on rational numbers is as follows . For x and y are both rational numbers .

x*y = x+y-xy .

(a) Prove that * is associative
(b) Determine if * is distributive with respect to addition .

The binary operation $\displaystyle *$ on rational numbers is as follows.

For rationals $\displaystyle x\text{ and }y\!:\;\;x*y \:=\: x+y-xy$

(a) Prove that * is associative
We want to show that: .$\displaystyle (a*b)*c \:=\:a*(b*c)$

$\displaystyle (a*b)*c \;=\;(a+b-ab)*c$

. . . . . . $\displaystyle =\;(a + b-ab) + c - (a+b-ab)c$

. . . . . . $\displaystyle =\; a + b - ab + c - ac - bc + abc$

. . . . . . $\displaystyle = \;a + b + c -(ab + bc + ac) + abc$ .[1]

$\displaystyle a*(b*c) \;=\;a*(b + c - bc)$

. . . . . . $\displaystyle = \;a + (b + c - bc) - a(b + c - bc)$

. . . . . . $\displaystyle =\;a + b + c + -bc - ab - ac + abc$

. . . . . . $\displaystyle = \;a + b + c - (ab + bc + ac) + abc$ .[2]

Since [1] = [2], the operation is associative.

(b) Determine if * is distributive with respect to addition .
Does $\displaystyle a*(b+c) \:=\:(a*b) + (a*c)$ ?

$\displaystyle a*(b+c) \;=\;a + (b+c) - a(b+c)$

. . . . . . .$\displaystyle = \;a + b + c - ab - ac$ .[3]

$\displaystyle (a*b) + (a*c) \;=\;(a + b - ab) + (a + c - ac)$

. . . . . . .$\displaystyle = \;{\color{red}2}a + b + c - ab - ac$ .[4]

Since [3] $\displaystyle \neq$ [4], the operation is not distributive over addition.