We explained the inclusion-exclusion formula (for 2 sets, 3 sets, to n sets) and showed some applications. The last part on Euler function will be skipped in this course (although I just added it in this year...).
The formula is used to compute the size of the union. In order to do this, for every element in the union, we want the formula to count it exactly once (not zero times, not two times, etc), and for every element not in the union, we want the formula not to count it. The latter part is clear, and so we focus on the former part. To prove the former part, we divide into different cases, e.g. whether the element belongs to one set, two sets, ..., n sets. For each case, we look at the formula and see its contribution to the sum. And this contribution is exactly the equation with the binomial coefficients, whose sum is equal to one (this proof is based on the binomial theorem). This means that each element in the union contributes exactly one to the formula, and so the formula computes the size of the union correctly. Please ask more specifically if you find it is still not clear. You can also come to my office hour on Tuesday.
Sorry, I don't understand the proof of inclusion-exculsion formula for n sets
ReplyDeleteIn the proof , it wants to prove that one element only count once by the formula.
But the formula is used to count size of set, why it can used to count how many times des the element counts?
Thanks for your feedback. I will try to explain this on the tutorial.
ReplyDeleteThe formula is used to compute the size of the union. In order to do this, for every element in the union, we want the formula to count it exactly once (not zero times, not two times, etc), and for every element not in the union, we want the formula not to count it. The latter part is clear, and so we focus on the former part. To prove the former part, we divide into different cases, e.g. whether the element belongs to one set, two sets, ..., n sets. For each case, we look at the formula and see its contribution to the sum. And this contribution is exactly the equation with the binomial coefficients, whose sum is equal to one (this proof is based on the binomial theorem). This means that each element in the union contributes exactly one to the formula, and so the formula computes the size of the union correctly. Please ask more specifically if you find it is still not clear. You can also come to my office hour on Tuesday.
ReplyDeletethank you
ReplyDeleteI understand now