Venn Diagrams
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Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.
What a Venn diagram encodes
Two-set inclusion-exclusion
|A \cup B| = |A| + |B| - |A \cap B|
The intersection is counted twice on the right, so subtract once.
Only-regions
|A \setminus B| = |A| - |A \cap B|
'Only A' is A minus the overlap.
Two-circle counting
Two-set inclusion-exclusion
|A \cup B| = |A| + |B| - |A \cap B|
The intersection is counted twice on the right, so subtract once.
Only-regions
|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.
Three-circle counting
Three-set union
|A \cup B \cup C| = \sum|A| - \sum|A \cap B| + |A \cap B \cap C|
Add singles, subtract pairs, re-add the triple.
Exactly two sets
\sum (|A \cap B|) - 3|A \cap B \cap C|
Sum of the three pairwise figures minus three times the triple.
Exactly one set
|A \cup B \cup C| - \text{exactly two} - \text{all three}
Strip the multi-set members from the union.