For a finite set A$$Z

$,$ define the sumset A$+$A

to be the set

A$+$A$=\{$a$+$b:a$,$b in A$\}.$

$($a$)[2$ points$]$ For the set A$=\{3,1,4\}$

$,$ write down the elements of A$+$A

$($be sure to omit duplicate elements$).$

$($b$)[6$ points$]$ Show that $|$A$+$A$|>=2|$A$|1$

$.$

$($c$)[2$ points$]$ Give an example of A$$Z

with $|$A$|>=5$

such that $|$A$+$A$|=2|$A$|1$