Let it be a non-empty finite set, a non-empty set is a subset family, and what satisfies it is called a subset family: it is a cover of a set.

How to calculate the order of a finite set by calculating the order of each subset in the cover:.

First, the principle of addition

If a subset family is not only a cover of a set, but also satisfies the following requirements

, ,

Then the cover is a partition of the finite set m.

addition principle

Let a nonempty finite set be a division of nonempty finite sets, then

Addition principle is a basic counting principle in combinatorial mathematics.

Second, the simple form of the principle of inclusion and exclusion.

What does it matter if it is not necessarily satisfied, that is to say, it may exist?

Theorem 1 (I)

Prove that if,,, then.

According to addition principle,,, So

Theorem 2 assumes that it is a subset of a set, then.

Proved by Morgan's Law and addition principle.

It is also derived from the theorem 1

Third, the general form of the principle of inclusion and exclusion.

Theorem 3.

certificate

From the theorem 1, the conclusion holds.

If the conclusion holds, then

.

The conclusion is established immediately.

According to the inductive principle, the conclusion holds for any positive integer.

morgan law

Sum formula (complete works)

Theorem 4 is set as a complete set, then

Also known as sieve formula, it is usually used to calculate the number of elements without any attributes.