Venn Diagrams
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What a Venn diagram encodes
Two-set union
|A \cup B| = |A| + |B| - |A \cap B|
The overlap is counted twice on the right, so subtract it once.
Only A
|A \setminus B| = |A| - |A \cap B|
Only A is A minus the overlap.
Number of parts
2^n
n circles make 2 to the power n parts, including the outside.
Two-circle counting
Two-set union
|A \cup B| = |A| + |B| - |A \cap B|
The overlap is counted twice on the right, so subtract it once.
Only A
|A \setminus B| = |A| - |A \cap B|
Only A is A minus the overlap.
Neither
\text{neither} = \text{total} - |A \cup B|
Everyone outside both circles.
Exactly one
|A| + |B| - 2|A \cap B|
Only A plus only B.
Three-circle counting
Three-set union
|A \cup B \cup C| = \Sigma|A| - \Sigma|A \cap B| + |A \cap B \cap C|
Add singles, subtract pairs, add the triple back.
Exactly two
\Sigma|A \cap B| - 3|A \cap B \cap C|
Sum of the pair figures minus three times the triple.
At least two
\Sigma|A \cap B| - 2|A \cap B \cap C|
Exactly two plus the triple.
Exactly one
|A \cup B \cup C| - \text{exactly two} - \text{all three}
Remove the multi-group members from the union.