abstract algebra Let x be an operation on A x elementof Bx1

abstract algebra

Let x be an operation on A = {x elementof B|x-1} defined by, x+y = x+y +xy prove this is a group Find Closure Associating Identity element prove formal proof that O is an identity) Identity Inverse find the inverse of x lessthanorequalto A

Solution

a) Closure : Let x,y in *

x * y = x + y + xy is not equal to -1

b) Associativity : Let x, y in *

x * y = x + y + xy = y + x + yx = y * x

c) Identity Element : x * 0 = x + 0 + 0x = x + 0 = x = 0*x

Hence 0 is identity

d) Identity Inverse : x * x^-1 = x + x^-1 + x^-1x = 0 => x + x^-1 + 1 = 0 => x^-1 = - (x+1)

Also if -(x+1) = -1 => -x-1 = -1 => -x = 0 => x=0 which is not possible

Hence -(x+1) lies in group