I try to understand and figure out to answer.

1) Here is an example of a set consisting of two element $\displaystyle G$ := {a,e} such that ae = a and ee = e (e is a "right identity") and each element has a left inverse ee=e and ea=e. Also set aa = a. Is the multiplication defined on $\displaystyle G$ associative? Is $\displaystyle G$ a group? justify your answer.

a) (aa)a = a(aa) -> aa = aa -> a=a

b) (ee)e = e(ee) -> ee = ee -> e=e

c) (aa)a = a(ae) -> aa = ae -> a=a ? (i am not sure)

d) (ae)e = a(ea) -> ae = ea -> e=e ?

e) (ae)e = a(ee) -> ae = ee -> e=e ?

2) Show that if $\displaystyle G$ is any set with an associative multiplication such that $\displaystyle G$ contains a right identity.

I put attach with word because there is equation. Thanks