STEP-BY-STEP LESSON · §9.3
Counting Elements of Disjoint Sets: The Addition Rule
Add cases without double counting.
Before you begin
Sets and membership
A set records which objects belong to it. Repetition and listing order do not change a set. x∈A says that x is an element; A⊆B says every element of A is also in B.
Intersection
A∩B contains objects in both A and B. In probability this is the event that both conditions occur. Count each common outcome once.
Symbols
- |A∪B|
- size of a union
- |A∩B|
- size of the overlap
- A\B
- Elements of A that do not belong to B
Definitions and notation for this topic
Sum rule
Adds the numbers of possibilities in mutually exclusive cases.
An option must not be counted twice.
Separate glossary example
Choose one of 3 red tickets or one of 2 blue tickets: 5 choices.
Inclusion–exclusion
|A∪B|=|A|+|B|−|A∩B|
For two finite sets A and B, add their sizes and subtract the overlap once to obtain the size of their union.
This formula applies to two finite sets.
Separate glossary example
|A∪B|=|A|+|B|−|A∩B|
Step by step
Step 1 / 6
Define cases
Separate results into sets A and B. Check whether one result can be in both.
Overlap determines the counting formula.
Worked example
In a group, 8 study Python, 6 Java, and 3 both. How many study at least one?
- Let A be the Python group and B the Java group; at least one asks for A∪B.
- The sum 8+6=14 includes three students twice.
- Subtract 3: 14−3=11.
Optional self-check
Of 20 people, 9 like tea, 8 coffee, and 3 both. How many like exactly one, and how many neither?
Show answer
Exactly one: (9−3)+(8−3)=11. At least one: 9+8−3=14, so neither: 20−14=6.