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: $$$card(A \times B)=card(A) \times card(B)=3 \cdot 5 =15 $$$