Equivalent Sets: Definition & Example

Equivalent Sets: Definition & Example
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  • 0:05 Definition of a Set
  • 0:52 Equal and Equivalent Sets
  • 2:54 Notation and Cardinality
  • 4:11 Lesson Summary
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Lesson Transcript
Instructor: Karin Gonzalez

Karin has taught middle and high school Health and has a master's degree in social work.

In this lesson, you'll learn the definition of equivalent sets. We'll look at some properties and terms related to equivalent sets, as well as examples so you can gain a better understanding of this concept.

Definition of a Set

Before we get into the definition of an equivalent set, we need to first know what a set is. A set is a collection of elements that are usually related. They are indicated with brackets: { }. We can have a set containing numbers, words, or even pictures. Here are some examples of sets:

  • {January, March, May, November}
  • {1, 2, 3, 4, 5, 6}

When a set continues on for infinity, the last element in the set is followed by three dots known as an ellipsis, which indicates that the numbers continue. An example is shown here: {1, 2, 3, 4, 5, 6. . . }.

Equal and Equivalent Sets

When we have two sets that have the exact same elements, we call them equal sets. It does not matter what order the elements are in. It just matters that the same elements are in each set. Here are some examples of equal sets:

  • {1, 3, 5, 7} and {7, 5, 3, 1}
  • {January, March, May, November} and {May, March, January, November}

An equivalent set is simply a set with an equal number of elements. The sets do not have to have the same exact elements, just the same number of elements. Let's take a look at some examples:

Example 1

  • Set A: {A, B, C, D, E}
  • Set B: {January, February, March, April, May}

Even though Sets A and B have completely different elements (Set A comprises letters, and Set B comprises months of the year), they have the same amount of elements, which is five. Set A contains five letters and Set B contains five months. That makes them equivalent sets!

Example 2

  • Set C: {Sweater, Mittens, Scarf, Jacket}
  • Set D: {Apples, Bananas, Peaches, Grapes}

Set C and Set D both comprise word elements in completely different categories (Set C comprises articles of clothing you would wear when cold, and Set D comprises types of fruit), but they both have the same amount of elements, which is four. That makes them equivalent sets!

Example 3

Example 3 of Equivalent Sets
Equivalent Sets E and F.

We use a picture in this example to illustrate that sets don't have to contain elements that are letters, numbers, or words. Some sets contain images. In this case, Set E contains three faces. It is still equivalent to Set F because it has the same number of elements.

Notation and Cardinality

When we speak of equivalence of sets, we use the equivalent sign, which is the tilde (~) sign. So if we wanted to say that set C was equivalent to set D, we would write: Set C ~ Set D.

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