# 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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Instructor
Karin Gonzalez

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

Expert Contributor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

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

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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## Exploring Relationships Between Equal and Equivalent Sets Using Real-World Examples

### Reminders:

• Two sets are equal if they contain the same elements.
• Two sets are equivalent if they have the same cardinality or the same number of elements.

### Questions:

1. Jamie and Grace go shopping. Jamie buys 2 shirts, 1 pair of shoes, and 3 pairs of pants. Grace buys 1 pair of shoes, 1 shirt, 1 pair of pants, 1 necklace, 1 ring, and 1 bottle of perfume. Is the set of things that Jamie bought equal to the set of things that Grace bought? Why or why not? Are the two sets equivalent? Why or why not?
2. Laura and Tristyn both have \$4. Laura has 4 one-dollar bills. Tristyn has 1 two-dollar bill, 1 one-dollar bill, and 2 fifty-cent coins. Is Laura's set of currency equal to Tristyn's set of currency? Why or why not? Are the two sets equivalent? Why or why not?
3. Erica and Tessa are both given a swag bag at a fashion show they are attending. Each bag contains a tube of mascara, a candy cane, a tube of lip gloss, and a bottle of hand lotion. Is the set of contents of Erica's bag equal to the set of contents of Tessa's bag? Why or why not? Are they equivalent sets? Why or why not?
4. Based on your answers to these questions, do you think all equal sets are also equivalent sets? Explain.
5. Based on your answers to these questions, do you think that all equivalent sets are also equal sets? Explain.

### Solutions:

1. The set of things that Jamie bought is not equal to the set of things that Grace bought, because they bought different things, so the sets contain different elements. However, the set of things that Jamie bought is equivalent to the set of things that Grace bought, because they each bought 6 things total, so the two sets each have the same number of things in them.
2. The set of Laura's currency is not equal to the set of Tristyn's currency, because they have different types of currency that add up to \$4. However, Laura's set of currency consists of 4 pieces of currency, and Tristyn's set of currency also consists of 4 pieces of currency, so the two sets are equivalent since they contain the same number of elements.
3. Since the swag bags have the exact same contents, the set of contents of Erica's bag is equal to the set of contents of Tessa's bag, because they contain the exact same elements. They are also equivalent sets because they both contain 4 items, so they have the same number of elements.
4. Yes, all equal sets are also equivalent sets. Equal sets have the exact same elements, so they must have the same number of elements. Therefore, equal sets must also be equivalent.
5. No, not all equivalent sets are also equal sets. We saw that this is the case with the first two questions because we had sets that are equivalent, but not equal.

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