Answer
a- $\left \{ 1,2,3,4,9,16 \right \}$
b-$\varnothing$
c- $A_{1},
A_{2},
A_{3}\, and\,
A_{4}$ are not mutually disjoint
Work Step by Step
$A_{i}=\left \{ i,i^{2} \right \}\,\,for\,\, i=1,2,3,4$$A_{1}= \left \{ 1,1 \right \},A_{2}=\left \{ 2,4 \right \},A_{3}=\left \{ 3,9 \right \},A_{4}=\left \{ 4,16 \right \}$
$a.\,\, \,A_{1}\cup A_{2}\cup A_{3}\cup A_{4}=\left \{ 1,1 \right \}\cup \left \{ 2,4 \right \}\cup \left \{ 3,9 \right \}\cup \left \{ 4,16 \right \}=\left \{ 1,2,3,4,9,16 \right \}$
$b. \,\, A_{1}\cap A_{2}\cap A_{3}\cap A_{4}= \left \{ 1,1 \right \}\cap \left \{ 2,4 \right \}\cap \left \{ 3,9 \right \}\cap \left \{ 4,16 \right \}= \varnothing $
$c.\,\, A_{1},A_{2},A_{3}\,and A_{4}\,\, are\,not\,mutually\,\, disjoint \, because \,\, \, A_{2}\,, A_{4}\, \,both\,\, contain \,\,4\,\, A_{2}\cap A_{4}=\left \{ 2,4 \right \}\cap \left \{ 4,16 \right \}=\left \{ 4 \right \}\neq \varnothing $