![]() Consider an element which belongs to exactly k sets, say A1, A2, A3, …, Ak. Inclusion-Exclusion (n sets) |A1[ A2[ A3[ … [ An| sum of sizes of all single sets – sum of sizes of all 2-set intersections + sum of sizes of all 3-set intersections – sum of sizes of all 4-set intersections … + (–1)n+1 × sum of sizes of intersections of all n sets We want to show that every element is counted exactly once. Inclusion-Exclusion (n sets) sum of sizes of all single sets – sum of sizes of all 2-set intersections + sum of sizes of all 3-set intersections – sum of sizes of all 4-set intersections … + (–1)n+1 × sum of sizes of intersections of all n sets Inclusion-Exclusion (n sets) What is the inclusion-exclusion formula for the union of n sets? Inclusion-Exclusion (3 sets) |A| 30 know Java 18 know C++ 26 know C# 9 know both Java and C++ 16 know both Java and C# 8 know both C++ and C# 47 know at least one language. It is clear that S is the union of A and B, but notice that A and B are not disjoint. Let B be the set of integers from 1 to 1000 that are multiples of 5. Let A be the set of integers from 1 to 1000 that are multiples of 3. Inclusion-Exclusion (2 sets) Let S be the set of integers from 1 through 1000 that are multiples of 3 or multiples of 5. Inclusion-Exclusion (2 sets) For two arbitrary sets A and B A B Sum Rule If sets A and B are disjoint, then |AB| = |A| + |B| A B ![]() Inclusion-Exclusion Principle Lecture 14: Oct 28
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