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Supplemental Math: Study Aid1 chapters | 19 lessons

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Lesson Transcript

Instructor:
*David Liano*

After completing this lesson, you will be able to define universal set and write examples of universal sets. You will also be able to utilize universal sets to create subsets of universal sets.

A universal set does not have to be the set of everything that is known or thought to exist - such as the planets, extraterrestrial life and the galaxies - even though that would be one example of a universal set. A **universal set** is all the elements, or members, of any group under consideration.

For instance, all the stars of the Milky Way galaxy could be a universal set if we are discussing all the stars of the Milky Way galaxy. This type of universal set might be appropriate for astronomers, but it is still a pretty large set of objects to consider.

A typical universal set in mathematics is the set of natural numbers as shown below: **N** = {1, 2, 3, 4, ...}.

Boldface capital letters are sometimes used to identify certain number sets, such as **N** for natural numbers. We usually use braces to enclose a set. The ellipsis mark (...) tells us that the set of natural numbers goes on without end; so this universal set is also an infinite set. However, universal sets can also be finite sets.

Let's look at an example of a universal set that is finite. The set of all the presidents of the United States is an example of a universal set that is finite. This set may increase every four years, but at any given time, it is a finite universal set if we are discussing all the men who have been president of the United States.

Sets are usually named with a capital letter. Therefore, the universal set is usually named with the capital letter **U**. This will be the notation used in this lesson.

Sometimes, alternate notation might be used for a universal set, such as the example of the set of natural numbers shown above. The set of natural numbers is not necessarily a universal set. Whether a set is a universal set is based on the structure of a problem or on the situation under examination. But the point here is that alternate notation can be used to name a universal set as long as it is practical and clear to the observer.

If all the elements of set **A** are also elements of set **B**, then **A** is a **subset** of **B**. This means that subsets can be created from any defined universal set. We should first acknowledge that any universal set is a subset of itself. However, a subset usually has less elements than the universal set from which it is created.

Let's go back to the set of natural numbers. Suppose we wanted to list all the natural numbers that satisfy the equation 20 < *x* < 25. This subset is shown next: {21, 22, 23, 24}.

The set of natural numbers itself is a subset of the set of real numbers, which could be another example of a universal set. Again, what is or isn't a universal set is based on the context of a problem.

Let's next go back to the universal set we created for the presidents of the United States: **U** = the set of the presidents of the United States. We want to create a subset from **U**.

We will create a subset of all the presidents of the United States who died in office. Let's call this set **A**, which is defined next: **A** = {Harrison, Taylor, Lincoln, Garfield, McKinley, Harding, F. Roosevelt, Kennedy}.

The inside of the rectangular shape represents the universal set. In other words, all the elements, or members, of **U** are represented by the area inside the rectangle. The subset **A** is represented by the circle. Of course, the total area of the circle needs to be inside the area of the rectangle.

Let's show one more example of a universal set and a subset of a universal set using a topic in mathematics. This will give us a chance to show additional notation used in sets. Our universal set will be all positive odd numbers less than 100: **U** = {1, 3, 5, 7, ... 99}.

The above notation shows that the pattern of consecutive positive odd numbers will continue to the number 99. Our subset will be all the prime numbers that are elements of this universal set, and we will call it set **B**: **B** = {3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.

This notation can be read as the set of all numbers *x* that are elements of the universal set, such that *x* is a prime number. The curvy E-symbol states that *x* is an element of **U**, and the vertical line (|) is understood to mean 'such that.'

A **universal set** is the set of all elements, or members, of a group under consideration. This group is usually relevant to a particular situation, such as a mathematical problem or some point of discussion. The creation of universal sets is handy when discussing a specific issue that pertains to only a certain set of members, such as the senior class of a high school. Finally, additional sets, called subsets, can be created from a universal set.

By the end of this lesson you should be able to define, interpret, and create a universal set and a related subset.

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Supplemental Math: Study Aid1 chapters | 19 lessons

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