# binary operation

Prove that any non empty set G with a binary operation satisfying the following requirements is a group:

a) if x, y in G then x.y in G

b) for all x,y,z in G, (x.y).z = x.(y.z)

c) for all a,b in G, the equations x.a =b and a.y =b both have solutions.

Hint: 1st show that if a in G then a has a right and left identity and be sure to show they are the same.

2ed show that the identity in step #1 works for any b in G

Note: 1st and 2ed steps can be reversed

3ed show that for any a in G a has a left and right inverse and they are the same.

https://brainmass.com/math/discrete-math/discrete-math-binary-operation-346348

#### Solution Preview

Dear Student

Given that for all a,b in G, the equations x.a =b and a.y =b both have solutions, hence we should have a solution for the equation x.a=a too.

This implies that there is an element 'e' (the solution for the above equation for x=e) such that e.a=a, ...

#### Solution Summary

This solution shows how to prove that any non empty set G with a binary operation, satisfying the following requirements is a group.