The principle of addition and multplication
Calculate the cardinal of $A \cup B$ and $A \times B$ if $A=\{1,3,4,5,6,7\}$ and $B=\{3,4,7,8,9\}$.
First the cardinal ones are calculated of $A$ and $B$: $$card(A)=6$$ $$card(B)=5$$ Next the intersection of both sets is calculated: $$A\cap B=\lbrace3,4,7\rbrace$$ and later its cardinal one: $$card(A\cap B)=3$$ Finally, to calculate $$card(A\cup B)$$ we apply the principle of addition, and to calculate the cardinal one of $A\times B$ we apply the principle of multiplication $$card(A\cup B)=card(A)+card(B)-card(A\cap B)=6+5-3=8$$ $$card(A)\times card(B)=6\cdot5=30$$
$card(A\cup B)=8$; $card(A)\times card(B)=30$