# Examples of union of sets

To understand what the **union of sets** is, you must first distinguish what a **set is** ; which is a **collection of elements** that can be recognized or distinguished and; that share **one or more characteristics** .

According to the **theory of sets,** the study of the **properties and relationships** between different sets, have among the main promoters of these theories, **B. Bolzano (Prague, 1848) and G. Cantor ****(Russia, 1845)** who laid the foundations of the **modern mathematics** . Although, they had many improvements during the twentieth century at the initiative of other students of mathematics, including **Nicolás Bourbaki** .

The **union of sets** is a **new set** , which is obtained by grouping two or more sets; resulting in the new collection of objects; which has as **elements** those that were previously, in each set that has been used within the grouping operation.

Each **set** must have its elements well defined. In **mathematics** we can take as **examples ****of sets** :

- All
**even numbers**greater than 1 and less than 15, this means that the set will be made up of elements 2, 4, 6, 8, 10, 12 and 14. - The
**integers**that are a solution of the equation**X**; in this case, its elements would be -2 and 2.^{2}-4 = 0

**Set notation and union of sets**

When establishing **sets** , the word **elements or members is used** to designate each object in the set and; characters such as **{}** (braces) are used; within which are included the **elements separated by commas or using a quality** .

That is, it can occur in two ways: detailing each element, it is said to be expressed **by extension** or; simply indicating the **condition** that defines the elements, in a univocal way, then it is said that they are given **by understanding** .

There is also a **graphical representation** of the whole, which is known as diagrams Venn; in which all the **elements of the set** are enclosed within shapes, using circles or rectangles.

These **diagrams** can also show the **union of sets; **containing multiple collections of items. Furthermore, it should be mentioned that the **sets** can be **finite or infinite; **where the union is represented by the letter U, using the **notation ****AUB** and that as a set is expressed as follows:

**AUB** = **{** X / X is an element of A or X is an element of B **}**

An example, given two **sets A and B** , Venn diagrams for the **union of sets** can be of 3 forms, depending on whether they have some elements in common (part of the intersection); no element in common or; one set included within the other. Let’s see an example that illustrates it:

Another example,

# Other examples of union of sets

- Let A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}, finite sets

The result being **AUB** = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

- Dice A = {oranges, tangerines, apples} and B = {bananas, watermelons, cherries};

Then **AUB** = {oranges, tangerines, apples, bananas, watermelons, cherries}

- Given X = {cats, cougars, lions} and Z = {gazelles, buffaloes, giraffes};
**XUZ**= {cats, cougars, lions, gazelles, buffaloes, giraffes} - If P = {X / X is a person walking} and Q = {X // X is a person running}

Then **PUQ =** {X / X is a person walking or running}

- A = {dogs, cats, parrots} and B = {guinea pigs, canaries, turtles};

**AUB** = {dogs, cats, parrots, guinea pigs, canaries, turtles}

- M = {X / X is an even natural number}; O = {X / X is an odd natural number}, then
**MUO**= IN, where IN is the**infinite set**of natural numbers.