Disjoint sets


Disjoint sets, or incompatible sets, are those that do not have any element in common. That is, the sets M and N are disjoint if their intersection is an empty set.

In other words, the sets M and N are disjoint if none of the elements of the first is in the second and vice versa. In formal terms, this can be expressed as follows:

In the expression above, xi is any of the elements contained in the set N. While xj is any of the elements of the set M.

Likewise, as we mentioned previously, two sets M and N are disjoint if their intersection is an empty set, as seen in the following expression:

We can conclude, then, that disjoint sets are mutually exclusive. This is because when an element belongs to M, for the same reason, it cannot be part of N and vice versa.

In the following image, we can observe two disjoint sets in a Venn diagram:

Examples of disjoint sets

Some examples are the following:

  • Even numbers greater than 25 and odd numbers less than 24.
  • People who live in the city of Madrid and people who live in Mexico City, on the same day and at the same time.
  • People who voted for party x in the 2016 Peruvian presidential elections and people who voted for party and in those elections.

Paired disjoint sets

A group of (more than two) sets will be disjoint by pairs or mutually disjoint if, when taking any two sets from the collective, they are always disjoint.

That is, in formal terms, we would have the following, where Ni and Nj belong to a family of sets that are disjoint by pairs:

It should be noted that a family of sets is the grouping of several sets.

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