Show that the binary operation * on A = R – {-1} defined as a*b = a + b + ab for all a,b belongs to A is commutative and associative on A. also find the identity element of  * in A and prove that every element of A in invertible

mention each and every formula and minute details

Asked by haroonrashidgkp | 7th Jun, 2018, 11:03: AM

Expert Answer:

begin mathsize 16px style straight a comma space straight b element of space straight R space rightwards double arrow straight a space plus space straight b space element of space straight R space and space ab space element of space straight R
rightwards double arrow straight a space plus space straight b space plus space ab space element of space straight R
Hence comma space asterisk times space is space straight a space binary space operation space on space straight R
Commutativity colon
For space all space straight a comma space straight b space element of space straight R
straight a space asterisk times space straight b space equals space straight a space plus space straight b space plus space ab
space space space space space space space space space space space equals space straight a space plus space straight b space minus space ba space space space space space space space space space space space commutativity space of space addition space and space multiplication
space space space space space space space space space space space equals space straight b space asterisk times space straight a
straight a asterisk times space straight b space equals space space straight b space asterisk times space straight a
asterisk times space is space straight a space commutative space on space straight R.
ASSociativity space colon space
open parentheses straight a asterisk times straight b close parentheses asterisk times straight c space equals space open parentheses straight a space plus space straight b space plus space ab close parentheses asterisk times straight c space
space space space space space space space space space space space space space space space space space equals straight a space plus space straight b space plus space ab space plus space straight c space plus space open parentheses straight a space plus space straight b space plus space ab close parentheses straight c
space space space space space space space space space space space space space space space space space equals straight a space plus space straight b space plus space ab space plus space straight c space plus space ac space plus space bc space plus space abc
Similarly space you space can space solve space for space straight a asterisk times open parentheses straight b asterisk times straight c close parentheses space we space get space
open parentheses straight a asterisk times straight b close parentheses asterisk times straight c space equals space straight a asterisk times open parentheses straight b asterisk times straight c close parentheses
asterisk times space is space straight a space associative space on space straight R.

Identity space element space colon space Let space straight e space be space the space identity space element space in space straight R comma
straight a asterisk times straight e space equals space straight a space equals space straight e asterisk times straight a space space space for space all space straight a element of straight R
straight a asterisk times straight e space space equals space straight a
straight a space plus space straight e space plus space ae space equals space straight a
straight e left parenthesis 1 space plus space straight a right parenthesis space equals space 0
straight e space equals space 0 space which space is space identity space element space in space straight R.
Inverse space of space an space element
Let space straight a space be space the space arbitary space element space of space straight R space and space straight b space be space the space inverse space of space straight a.
straight a asterisk times straight b equals 0 equals straight b asterisk times straight a
straight a asterisk times straight b equals 0
straight a space plus space straight b space plus space ab space equals space 0
straight b left parenthesis 1 space plus straight a right parenthesis space equals space minus straight a
straight b space equals space fraction numerator negative straight a over denominator 1 plus straight a end fraction
Inverse space exist space only space if
straight a element of straight R space minus space left curly bracket negative 1 right curly bracket
So comma space every space element space of space straight R space is space invertible space except space minus 1.
Every space element space straight a open parentheses not equal to negative 1 space close parentheses element of straight R space has space its space inerse space fraction numerator negative straight a over denominator 1 plus straight a end fraction. end style

Answered by Sneha shidid | 8th Jun, 2018, 12:09: PM