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.

Example

If the sets are: $A = {a, b, c, d, e}$ and $B = {x, y, z}$ , then: $c a r d (A) = 5 c a r d (B) = 3$ and therefore: $c a r d (A \cup B) = c a r d (A) + c a r d (B) = 5 + 3 = 8$ 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.

Example

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: $c a r d (A) = 5 c a r d (C) = 4 c a r d (A \cap C) = 2$ and therefore: $c a r d (A \cup C) = c a r d (A) + c a r d (C) - c a r d (A \cap C) =$ $= 5 + 4 - 2 = 7$

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.

Example

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