1 December 2023 The union of sets is a fundamental term in set theory in the area of ​​mathematics and is used to describe the combination of elements of two or more sets. In this article we will explore the definition of the union of sets, the properties that make it unique, and examples of how it is used in mathematics.

## Meaning of union of sets in statistics and mathematics

The union of sets is an operation used in mathematics and statistics that allows two or more sets to be combined into one. In simple terms, the union of sets represents the combination of elements from different sets in one set. This operation is denoted by the symbol ‘‘ and is read as ‘union’.

In statistics, the union of sets is used to find the probability that at least one of two or more events will occur. For example, if you have two sets A and B, the probability that at least one of them occurs can be found by the union of the two sets and the union probability formula.

In mathematics, the union of sets is used to find the combination of elements of two or more sets and create a new set without duplicates. The union of sets also is useful for finding the plugin of a set, which is the set of elements that do not belong to the original set.

## What is the difference between the union and the intersection of sets?

Union and intersection are two fundamental operations in set theory, which are used to combine sets and obtain new ones. The main difference between union and intersection is that the union of two sets contains all the elements of both sets, while the intersection of two sets contains only the elements that are present in both sets.

More specifically, if ‘A’ and ‘B’ are two sets, the union of ‘A’ and ‘B’ (represented by A ∪ B) is a set containing all the elements belonging to A or B or both. On the other hand, the intersection of ‘A’ and ‘B’ (represented by A ∩ B) is a set containing all elements belonging to both A and B.

In visual terms, if we imagine two sets as two overlapping circles, the union of the two sets would be the circle that contains all the elements of both sets, including the overlapping areas, while The intersection would be the overlapping area common to both circles.

## What are the properties of the union of sets?

The union of sets has several important properties that They are frequently used in set theory. and in various mathematical and statistical applications. Some of the most common properties of the union of sets are described below:

### cardinality

The cardinality of a set is the number of elements that a set contains. The cardinality is denoted by the symbol |a|, where ‘A’ is a set. For example, if A = {1, 2, 3}, then the cardinality of ‘A’ is 3, which is written as |A| = 3. In the context of the union of sets, the cardinality of the union of two sets can be calculated using the following formula:

• |A ∪ B| = |A| + |B| – |A ∩ B|

This formula is known as the include-exclude rule, and can be used to determine the number of elements in the union of two sets, without counting the elements that are repeated in both sets. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then the intersection of ‘A’ and ‘B’ is {2, 3}, so |A ∩ B| = 2. The union of ‘A’ and ‘B’ is {1, 2, 3, 4}, so |A ∪ B| = 4. Now, substituting the known values ​​in the formula, we have:

• |A ∪ B| = |A| + |B| – |A ∩ B| 4 = 3 + 3 – 2

Therefore, the cardinality of the union of ‘A’ and ‘B’ is 4, as expected. Cardinality is an important property of sets, since it allows us to quantify and compare sets and relationships between them.

### idempotence

The idempotency property is one of the properties of the union of sets. This property states that the union of a set with itself does not change the original set, that is, A ∪ A = A.

In other words, if A is a set, the union of A with itself is equal to A, and this is because when joining two equal sets, no new items added to the original set, and therefore no change occurs.

For example, if A = {1, 2, 3}, then A ∪ A = {1, 2, 3} ∪ {1, 2, 3} = {1, 2, 3}. In this case, the union of ‘A’ with itself does not add a new member to ‘A’, since elements 1, 2 and 3 are already in the original set. The idempotency property It is useful in set theory. and in various mathematical and statistical applications. This property allows us to simplify expressions involving the union of sets and can be used to prove other properties of sets.

### associative property

The associative property is another of the properties of the union of sets. This property sets that the way sets are grouped in a union does not affect the final result. In other words, if we have three sets A, B, and C, then (A ∪ B) ∪ C = A ∪ (B ∪ C).

This property is important because it allows us to group sets in the way that is most convenient for performing operations on them. For example, if we have three sets A, B and C, we can join A and B first and then join the result with C, or we can join B and C first and then join the result with A. In both cases, the end result is the same set.

For example, yes A = {1, 2}, B = {2, 3} and C = {3, 4}so:

• (A ∪ B) ∪ C = ({1, 2} ∪ {2, 3}) ∪ {3, 4} = {1, 2, 3} ∪ {3, 4} = {1, 2, 3, 4 }
• A ∪ (B ∪ C) = {1, 2} ∪ ({2, 3} ∪ {3, 4}) = {1, 2} ∪ {2, 3, 4} = {1, 2, 3, 4 }

In both cases, the final result is the set {1, 2, 3, 4}. This shows that the associative property is valid for the union of sets. The associative property also applies to other set operations, like intersection and difference.

### commutative property

The commutative property is another of the properties of the union of sets, and it establishes that the order in which two sets are joined does not affect the final result. In other words, if you have two sets A and B, then A ∪ B = B ∪ A. This property is important because it allows us to change the order in which the sets are joined in an expression without affecting the final result. For example, if we have two sets A = {1, 2} and B = {2, 3}, then:

• A ∪ B = {1, 2} ∪ {2, 3} = {1, 2, 3}
• B ∪ A = {2, 3} ∪ {1, 2} = {1, 2, 3}

In both cases, the final result is the set {1, 2, 3}. This shows that the commutative property is valid for the union of sets. The commutative property also applies to other set operations, such as intersection and difference. This property It is useful for simplifying expressions. and to prove other properties of sets.

## Examples of the union of sets in mathematics

A simple example of the union of sets is the combination of two sets, A = {1,2,3} and B = {3,4,5}, to create a third set C = A ∪ B = {1,2,3,4,5}. In this example, set C contains all the elements of A and B with no duplicates.

Another example is the use of the union of sets to find the subset of a larger set. If we have a set U = {1,2,3,4,5} and two subsets A = {1,2,3} and B = {3,4,5}, then the union of the two subsetsA ∪ B = {1,2,3,4,5}, yields the complete set U.

Set Theory is a fundamental part of modern mathematics, and is used in various fields such as logic, computer theory, statistics, physics, among others. It’s a very extensive theory. and it is widely documented in sources such as Wikipedia and other reference books.