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High School Algebra I: Homework Help Resource25 chapters | 271 lessons

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Lesson Transcript

Instructor:
*Peter Kosek*

Peter has taught Mathematics at the college level and has a master's degree in Mathematics.

How can we relate two things? Is there a formal way to say that two things are equivalent, while also saying that two things are not equivalent? Equivalence relations can do that!

Suppose somebody was to say that raspberries are equivalent to strawberries. At first, you might be confused. How are they equivalent? Are blueberries also equivalent to strawberries?

You don't know! But, if someone was to explain that two berries are equivalent if they are the same color, you'd immediately understand why raspberries are equivalent to strawberries and that blueberries are not equivalent to strawberries. Is this an acceptable way to define equivalence?

Let's look at another example. What if someone was to say that two words, each longer than three letters, are equivalent if their first three letters are exactly the same? For example, robotics would be equivalent to robbery, and congress is equivalent to constant. Is this an acceptable way to define equivalence?

For our final example, what if someone was to say that two natural numbers are equivalent if they have share a common factor greater than one? For example, 6 is equivalent to 8, and 3 is equivalent to 18. Is this an acceptable way to define equivalence?

According to the mathematical way of defining equivalence, two of the three above examples are acceptable and one is not. Can you guess which one?

First off, let's describe a relation. A **relation** is the method by which we compare two elements in the same set. In our first example, the relation is having the same color. In our second example, our relation is having the same first three letters.

In order for our relation to be an acceptable way of defining equivalence, the relation between the elements must satisfy the following three criteria:

**1. Reflexive**: An element, *a*, is equivalent to itself

**2. Symmetric**: If *a* is equivalent to *b*, then *b* is equivalent to *a*

**3. Transitive**: If *a* is equivalent to *b*, and *b* is equivalent to *c*, then *a* is equivalent to *c*

Any relation satisfying these three conditions is called an **equivalence relation**.

Now that we have a formal way to define an equivalence relation, let's go back to our initial three attempts to define equivalence and see which one is not an equivalence relation.

**Example One: Are Raspberries Equivalent to Strawberries?**

Consider our first relation, 'having the same color', with the set we're considering being the set of all berries. First, we'll check if the relation is **reflexive**. Does a berry have the same color as itself? Yeah, that seems right. Good! This relation is reflexive.

Next, we'll check if the relation is **symmetric**. If one berry has the same color as another berry, does this second berry have the same color as the first berry? Again, that seems pretty obvious. Great! The relation is symmetric.

Lastly, we'll check if the relation is **transitive**. If one berry has the same color as a second berry and the second berry has the same color as a third berry, does the first berry and the third berry have the same color? Again, this seems to be true. Good! Our relation is transitive. Since our relation is reflexive, symmetric, and transitive, our relation is an **equivalence relation**!

**Example Two: Words with the Same Three Letters**

Let's take a look at the second relation, 'having the same first three letters', with the set we're considering including all words with three or more letters. First, we'll check if the relation is **reflexive**. Does a word have the same first three letters as itself? Yeah, that seems fairly obvious. Fantastic! This relation is reflexive.

Next, we'll check if the relation is **symmetric**. If one word has the same first three letters as another word, does this second word have the same first three letters as the first word? Again, that's true, so the relation is symmetric.

Lastly, we'll check if the relation is **transitive**. If one word has the same first three letters as a second word, and the second word has the same first three letters as a third word, does the first word and the third word have the same first three letters? Again, this seems to be true. Wonderful! Our relation is transitive. Since our relation is reflexive, symmetric, and transitive, our relation is an **equivalence relation**!

**Example Three: Natural Numbers**

Since I said only two of the first three examples were equivalence relations, we now know that the last example is not an equivalence relation. But why is that? Which of the three conditions was not met? We can check the **reflexive** and **symmetric** conditions and see that both of those hold. However, the culprit is the transitive condition.

To properly show that this relation is not transitive, we need to create an example showing this. Let's consider the numbers 6, 16, and 9. We see that 16 and 6 share a common factor greater than 1, namely 2. Also, 6 and 9 share a common factor greater than 1, namely 3. However, 16 and 9 do not share a common factor greater than 1. Therefore, this relation is not transitive. Hence, it is not an equivalence relation.

To check if a relation is an **equivalence relation**, it must be reflexive, symmetric, and transitive. If any of these three fail to be true, the relation is not an equivalence relation.

Remember that to be **reflexive**, an element, *a*, is equivalent to itself; it is **symmetric** if *a* is equivalent to *b*, then *b* is equivalent to *a*; and it is **transitive** if *a* is equivalent to *b*, and *b* is equivalent to *c*, then *a* is equivalent to *c*.

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High School Algebra I: Homework Help Resource25 chapters | 271 lessons

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