Set Notation: Definition & Examples

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  • 0:00 The Set Concept
  • 3:00 Sets of Number Systems
  • 3:58 Subsets
  • 4:30 Union and Intersection of Sets
  • 5:21 Cardinality of…
  • 6:12 Lesson Summary
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Lesson Transcript
Instructor
David Liano

David has a Master of Business Administration, a BS in Marketing, and a BA in History.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

After completing this lesson, you will know how to write a set using words and using mathematical symbols. You will also know how to define sets and how to interpret sets that are displayed in set notation.

The Set Concept

A set is simply a collection of items called elements, or members of the set. Each element is distinguishable from the other elements. Examples of sets are probably the best way to illustrate what a set is.

A set could be the countries of Europe. France would be an element of this set. The country of Argentina would not be an element of this set because it is located in South America. The city of Rome would not be an element of this set because it is a city in Europe, not a country in Europe. In the context of mathematics, a set could be all the integers greater than 10 and less than 20. The numbers 12 and 17 would be elements of this set, while the numbers 20 and 35 would not be elements of this set.

We can make any type of set we want. For instance, we could combine the sets above into one set. The elements in a set do not need to have any relationship except that they are elements of the same set. For instance, Babe Ruth and the number 1,000,000 could be in the same set. However, there is usually some connection between the elements of a set to make the set practical and useful.

Braces {} are usually used when writing down a set. It is often common to use capital letters to name a set. Let's say set A has the elements of 3, 5, and 7. We would write is as follows:

A = {3, 5, 7}

Set A has three elements. We can list the elements in any order, and we can list elements more than once. We can also write set A as follows:

A = {5, 3, 7}
A = {3, 5, 7, 7}

However, the elements have not changed, and there are still only three elements. We can also have sets within sets. That means that sets can be elements of other sets. Let's look at the following example:

B = { a, {a}, b, c, {d, e} }

In set B, there are five elements. Two of the elements are sets of letters. The other three elements are single letters. Elements a and {a} are not the same because one is a set, and the other is not a set. In addition, the letters d and e are not elements of set B, but the set {d, e} is an element of set B. This distinction between elements and sets is straightforward but often is a difficult rule to apply.

For sets with many elements, it might be helpful to abbreviate using the ellipse (. . .) symbol. For instance, the set containing all natural numbers from 1 to 199 would be cumbersome to write out completely. We could abbreviate as follows:

S = {1, 2, 3, … 199}

To show that an element is part of a set, we use a curvy E symbol. The number 5 is an element in set S, and this is shown in Figure 1 using the curvy E symbol (below).

Sets of Number Systems

We use certain letters to define various number systems. This helps to better define sets and to make them easier to write. We will use the following capital letters for the respective number system sets:

N = Natural Numbers {1, 2, 3, . . .}

Z = Integers {… -3, -2, -1, 0, 1, 2, 3, . . .}

Q = Rational Numbers

I = Irrational Numbers

R = Real Numbers

Let's take the set we mentioned earlier of natural numbers from 1 to 199. Using the letter N for natural numbers, we can write the set in set notation as shown in Figure 2 (below).

Interpret the vertical line after the variable x to mean 'such that.' Let's now write this set in words: set A consists of all elements x such that x is a natural number and x is less than 200.

Subsets

Let's consider two sets A and B shown below:

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Additional Activities

Extra Practice with Sets

The following examples will help you practice determining if sets are subsets of each other and practice finding the union and intersection of various sets.


Subsets

  • Let P be the set of all people who have at least one pet. Let D be the set of all people who have at least one dog. Is D a subset of P? Is P a subset of D?
  • Let A={1,2,3,4,5} and let B={1,3,5}. Is A a subset of B? Is B a subset of A?
  • Let A={1,2,3,4,5} and let B={1,2,3,4,5}. Is A a subset of B? Is B a subset of A?
  • Let R be the set of real numbers and let Z be the set of integers. Is R a subset of Z? Is Z a subset of R?


Solutions to Subset Examples

  • D is a subset of P since some of the people who have at least one pet also have at least one dog. P is not a subset of D, because there are people in P who are not in D (for example, maybe they only have a cat)
  • A is not a subset of B, because 2 and 4 are not elements of B. B is a subset of A since every element of B is also an element of A.
  • A is a subset of B and B is a subset of A since every element of A is an element of B and vice versa. When two sets are subsets of each other, the two sets are equal.
  • R is not a subset of Z, because there are some real numbers that are not integers (for example, 2.5). Z is a subset of R since every integer is a real number.


Union and Intersection

  • Let A={1,3,5,7} and B={2,4,6}. Find the union of A and B and the intersection of A and B.
  • Let A={1,2,3,4} and B={3,4,5,6}. Find the union of A and B and the intersection of A and B.


Solutions to Union and Intersection Examples

  • The union of A and B is {1,2,3,4,5,6,7} - these are all the elements that are in A or B or both. The intersection of A and B is the empty set { } since they have no elements in common.
  • The union of A and B is {1,2,3,4,5,6} - these are all the elements that are in A or B or both. The intersection of A and B is {3,4} - the elements they have in common.

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