Final answer:
The inclusion-exclusion principle states that for any two finite sets A and B, the cardinality of their union is equal to the sum of their individual cardinalities minus the cardinality of their intersection. This accounts for the overlap and ensures that common elements are not double-counted. The principle is expressed mathematically as |A ∪ B| = |A| + |B| - |A ∩ B|.
Step-by-step explanation:
Proof of the Inclusion-Exclusion Principle for Finite SetsWhen attempting to prove that for any two finite sets A and B, the size of their union |A ∪ B| is equal to the sum of the sizes of the individual sets minus the size of their intersection, i.e., |A ∪ B| = |A| + |B| - |A ∩ B|, you can visualize each set as containing a distinct number of elements with some possibly overlapping. These overlapping elements are counted in both sets but should only be counted once when determining the total number of elements in the union. The inclusion-exclusion principle accounts for this by subtracting the count of overlapping elements, ensuring they are not double-counted.
- Cardinality of the Union: When we take the union of sets A and B, we combine all the elements from both sets.
- Cardinality of the Intersection: The intersection of sets A and B contains elements that are common to both sets, which are counted in the sizes of both A and B.
- Correcting for Overlap: Subtracting the cardinality of the intersection from the combined sizes of A and B corrects for the double-counting of the elements that are in both A and B.
Therefore, |A ∪ B| = |A| + |B| - |A ∩ B| is a mathematical statement of the inclusion-exclusion principle for the specific case of two finite sets.