Sum Rule

INFO

For any disjoint sets $A$ and $B$,
$[A\cup B|=|A|+|B|]$
Keyword: OR

More Generally

To count the number of disjointed pairs:

• Count the number of choices in the first disjoint set
• Count the number of choices in the second disjoint set
• Add those two counts

CAUTION

This rule only applies when the sets are disjoint—i.e., they share no elements. If sets overlap, use the Principle of Inclusion-Exclusion instead.

Example

Suppose you have:

• 5 red shirts
• 7 blue shirts
  If you want to know how many shirts you can choose either red OR blue, the total is:

$$[5+7=12]$$

Edge Case

If sets are not disjoint, then:

$$[|A\cup B|=|A|+|B|-|A\cap B|]$$

This is a basic form of the Inclusion-Exclusion Principle.