GROUP

Question

does it satisfy for every subset of R?

Answer 1

every element in S will have unique identity and inverse,

for identity, e=0

x*0=x  { x+0+0*x = x  }

for inverse,

let y be inverse of x

x*y=0

x+y+xy=0

y=-x/(1+x)   { 1+x !=0)

x!= -1

Answer 2

Yes, S is the subgroup closed under * and it does show the abelian property. Remember its mentioned that s is improper subset of R, thus it should satisfy the property for the most.