00:01
To prove that the stat with the operation is a group, we need to verify these three axioms.
00:09
First, the operation is associated.
00:16
X operate y and then operates z equals x plus y plus x, operate z and this equals x plus y plus xy plus xy plus z plus x z plus x y plus y z plus x y plus x x this equals x plus y plus z plus y z plus y z plus x y plus x plus x plus x y z and you can check that these two expressions are the same and hence the operation is associated next, x operates the identity e equals x plus e plus xe, this equals x equals x, so the identity e equals 0.
01:43
And we can also check that e operates x also equals x.
01:53
The third x in x operate x inverse equals x plus x inverse plus x times x inverse and this equals e, which is zero.
02:10
So 1 plus x times x inverse equals minus x, which is subtract both sides by x.
02:28
And hence we get x inverse equals minus x over 1 plus x...