The Inclusion Exclusion Principle and Its More General Version

where U = {1, 2, . . . , n}. Example. Suppose we ask how many integers less than 120 are relatively prime to it. Since 120 has prime factors 2, 3, and 5, any number that is relatively prime to it is not a multiple of any of these. We can solve this using the Inclusion-Exclusion Principle as follows. Let A = {1, 2, . . . , 120}, the set of integers from 1 to 120. Let S1 be the multiples of 2 in A, S2 be the multiples of 3 in A, and S3 be the multiples of 5 in A. The set of numbers relatively prime to 120 are the numbers not in S1 ∪ S2 ∪ S3, so there are 120− |S1 ∪ S2 ∪ S3| many of them. By the theorem, ∣∣∣∣ 3 ⋃