The principle of addition and multplication

Next, two rules for counting the number of elements in sets.

• Principle of addition: To count the elements of the union of two sets that have no elements in common, just add together the cardinals in each set.

If the sets are: $A=\{ a,b,c,d,e \}$ and $B=\{x,y,z\}$, then: $$card(A)=5 \\ card(B)=3$$ and therefore: $$card(A \cup B) =card(A) + card(B) =5+3 =8$$ Nevertheless, if two sets have elements in common, the cardinals in each set will have to be added and the cardinal in the intersection will have to be reduced.

With the sets $A=\{ a,b,c,d,e \}$ and $C=\{ a,b,g,h \}$, the intersection of both (that is to say, the elements together) is $A \cap C = \{ a,b \} $. Then: $$card(A)=5 \\ card(C)=4 \\ card(A\cap C)=2$$ and therefore: $$card(A\cup C)=card(A)+card(C)-card(A \cap C)=$$ $$=5+4-2=7$$

• Principle of multiplication: To count the elements of the Cartesian product in two sets $A$ and $B$, it is necessary to multiply the cardinals of both sets.

If $A=\{ a,b,c,d,e \}$ and $B= \{ x,y,z \}$, then: $$card(A)=5 \\ card(B)=3$$ and therefore: $$card(A \times B)=card(A) \times card(B)=3 \cdot 5 =15 $$