$$31=n((A \times B) \cup (B \times A))=n(A \times B)+n(B \times A)-n((A \times B) \cap (B \times A))$$
$$=n(A).n(B)+n(B).n(A)-n((A \cap B)\times (B \cap A))$$
$$=2.n(A).n(B)-n(A \cap B).n(B \cap A)$$
$$=2 \times 4 \times 5-(n(A \cap B))^2$$
$$=40-(n(A \cap B))^2$$
$$ \Rightarrow (n(A \cap B))^2=40-31=9$$
$$ \Rightarrow n(A \cap B)= \sqrt{9} =3$$
$$ \Rightarrow n(A \cup B)=n(A)+n(B)-n(A \cap B)=4+5-3=6$$
$\Box$
