What the displayed Three-Set Inclusion–Exclusion means
Three-set inclusion–exclusion adds individual counts, subtracts pair overlaps, then restores the triple overlap.
Math calculator
Three-Set Inclusion–Exclusion Calculator
Find the size of a three-set union from individual and overlapping cardinalities. The displayed union cardinality includes enough working to inspect signs and scale.
Three-Set Inclusion–Exclusion inputs
Numerical setup
Auditing the overlap signs
Members in all three sets are added three times and subtracted three times, so the triple intersection must be restored once.
Testing Three-Set Inclusion–Exclusion beyond the example
Keep the Three-Set Inclusion–Exclusion ordering and membership rules attached to |A|. Read |B| under that same Three-Set Inclusion–Exclusion convention.
List a small nonempty Three-Set Inclusion–Exclusion example. When allowed, compare it with an empty Three-Set Inclusion–Exclusion case.
For a Three-Set Inclusion–Exclusion audit, retain |A| and |B|. Decide the likely direction of Union cardinality before rerunning Three-Set Inclusion–Exclusion. Change only |C|; the response in Union cardinality can then be traced within the Three-Set Inclusion–Exclusion setup.
Cross-checking Three-Set Inclusion–Exclusion
A boundary case can expose a Three-Set Inclusion–Exclusion setup error. Choose a simple |A|, preserve |B|, and predict Union cardinality. If the new Union cardinality conflicts with that prediction, inspect the Three-Set Inclusion–Exclusion entries before relying on the less convenient case.
Keep extra digits in Union cardinality until the next step is known. Early rounding can obscure whether |A| and |B| satisfy the Three-Set Inclusion–Exclusion relation. Round the final Union cardinality once, using precision appropriate to the original Three-Set Inclusion–Exclusion data.
Work backward from the displayed Union cardinality once. Ask whether |A| can produce that Union cardinality under |B|. If the reverse Three-Set Inclusion–Exclusion relationship fails, recheck the entry order and any convention attached to |C|.
A one-input trial is useful when Three-Set Inclusion–Exclusion behaves unexpectedly. Change |C| alone, retain |A| and |B|, and observe Union cardinality. Multiple simultaneous edits would make the cause of the changed Union cardinality ambiguous within the Three-Set Inclusion–Exclusion setup.
Reviewing Three-Set Inclusion–Exclusion in context
The three-set formula is a named case of the general inclusion-exclusion principle. For that connected step, see general inclusion-exclusion.
Questions about Three-Set Inclusion–Exclusion
What does Three-Set Inclusion–Exclusion calculate?
Three-set inclusion–exclusion adds individual counts, subtracts pair overlaps, then restores the triple overlap.
When is Three-Set Inclusion–Exclusion useful?
It counts unique survey respondents, customers, events, courses, skills, and records spread across three overlapping groups.
What can make Three-Set Inclusion–Exclusion misleading?
All supplied intersections must fit inside their component sets and describe one mutually consistent arrangement.
Checking Three-Set Inclusion–Exclusion independently
Mark how often each region was counted, then use + singles, − pairs, + triple.