Math

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 2 -4 = 0 ; in this case, its elements would be -2 and 2.

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

  1. 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}

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

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

  1. Given X = {cats, cougars, lions} and Z = {gazelles, buffaloes, giraffes}; XUZ = {cats, cougars, lions, gazelles, buffaloes, giraffes}
  2. 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}

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

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

  1. 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.

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