Q$7.$ Given functions $f_1,f_2,g_1,g_2$ such that $f_1(n)=O(g_1(n))$ and $f_2(n)=O(g_2(n)).$ For each of the

following statements, decide whether you think it is true or false and give a proof or

counterexample. $(12$ points$)$

$($a$)f_1(n)*f_2(n)=O(g_1(n)*g_2(n))$

$($b$)f_1(n)+f_2(n)=O(max(g_1(n),g_2(n)))$

$($c$)f_1(n{)}^2=O{g}_1(n{)}^2$

$($d$)log2f_1(n)=O(log2g_1(n))$