Prove G is a group

Question

Let G be a set with an operation * such that :
(1)G is closed under *
(2)* is associative
(3)There exists an element e in G such that e*x=x for all x in G
(4)Given x in G, there exists a y in G such that y*x=e

Prove that G is a group.(Thus you must show that x*e=x and x*y=e for e,y above)

? · Accepted Answer

There exists an element e in G such that e*x=x for all x in G…(1)
Given x in G,there exists a y in G such that y*x=e…(2)
Given y in G,there exists a z in G such that z*y=e…(3)
First x*y=e*(x*y);from (1)
=(z*y)*(x*y);from (3)
=z*(y*x)*y;associative property
=z*e*y
=z*(e*y);associative property
=z*y
=e ; from(3)
Second, x*e = x*(y*x) from (2)
=(x*y)*x; associative property
=e*x; from above
=x; from (1)

Prove G is a group

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