In Combinative, the principle of inclusion-exclusion of Moivre makes it possible to express the number of elements which satisfies one or the other properties given, according to the number of its elements which satisfy at the same time some of its properties.

It owes its name with the Mathématicien Abraham de Moivre.

Theorem

That is to say E a unit finished, whose each element can or not satisfy some of N properties $P\_1,\; \backslash \; ldots,\; P\_n$. Let us note $N\_\; \{i\_1,\; \backslash \; ldots,\; i\_k\}$ the number of elements of E which check the properties $P\_\; \{i\_1\},\; \backslash \; ldots,\; P\_\; \{i\_k\}$ (and possibly of other properties). Then the number NR of elements (distinct) of E which satisfy one or the other of these properties is given by

In an equivalent way, the number of elements of E which do not check any property is equal to

Particular case

Let us consider the particular case where N =2. That is to say E a finished unit, whose each element can or not satisfy one or the other of the two properties $P\_1$ and $P\_2$. Let us note

• $N\_\; \{1\}$ the number of elements of E which check $P\_1$,
• $N\_\; \{2\}$ the number of elements of E which check $P\_2$,
• $N\_\; \{1,2\}$ the number of elements of E which check at the same time $P\_1$ and $P\_\; \{2\}$.

Then the number NR of elements of E which satisfy one or the other of these properties is equal to

$N=N\_\; \{1\}\; +\; N\_\; \{2\}\; -\; N\_\; \{1,2\}$.

In other words, the number of objects checking one or the other of these two properties is equal to the sum of the numbers of objects checking each decreased property of the number of objects having at the same time the two properties.

Example

Among 20 students, 10 study mathematics, 11 study physics, and 4 study both. How much are there students who study neither mathematics nor physics?

To visualize we can build a Venn diagram.

We surround the elements which check the same property. E represents the whole group of students, M represents those which have the property “to study mathematics”, P represents those which has the property: “to study physics”.

We place in each part the number of students. Since four people study at the same time mathematics and physics, we defer one 4 in the intersection of the two circles. We must thus have 10-4=6 people who study mathematics but not physics and 11-4=7 people which studies physics but not mathematics. There thus remain 20 (6+4+7) =3 people who study neither mathematics nor physics.

This result is found easily by using the principle of inclusion-exclusion which gives the number of students not having these two properties 20-10-11+4=3.

Applications

The principle includes as inequalities, showing as the sum of the first terms of the formula is alternatively one raising and one undervaluing of the first member. These inequalities can be used whenever the complete formula is too cumbersome.

The principle of inclusion-exclusion is found in the Formule of the screen of Poincaré and the Formule of the screen of Poincaré of probability or in the Formule of inversion of Möbius.

Related article

• the Formula of the screen of Poincaré

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