Tag Archives: inclusion

PIE: proof by counting

Recall the setup: we have a universal set and a collection of subsets , , , and so on, up to . PIE claims that we can compute the number of elements of that are in none of the (that … Continue reading

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PIE: proof by algebra

In my previous post I stated a very formal, general form of the Principle of Inclusion-Exclusion, or PIE.1 In this post I am going to outline one proof of PIE. I’m not going to give a completely formal proof, because … Continue reading

Posted in combinatorics, induction, pattern, proof | Tagged , , , , , , | 2 Comments

Formal PIE

I’ve been talking informally about the Principle of Inclusion-Exclusion but I realized it would be useful to state it more formally before proceeding to some proofs. The only problem is that a fully formal statement of PIE has a lot … Continue reading

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Post without words #19

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Post without words #18

Posted in pattern, pictures, posts without words | Tagged , , , , , , | 5 Comments

PIE day

[This is part six in an ongoing series; previous posts can be found here: Differences of powers of consecutive integers, Differences of powers of consecutive integers, part II, Combinatorial proofs, Making our equation count, How to explain the principle of … Continue reading

Posted in combinatorics, counting | Tagged , , , , | 2 Comments

How to explain the principle of inclusion-exclusion?

I’ve been remiss in finishing my series of posts on a combinatorial proof. I still intend to, but I must confess that part of the reason I haven’t written for a while is that I’m sort of stuck. The next … Continue reading

Posted in combinatorics, meta | Tagged , , , | 7 Comments