Back To Course

Math 97: Introduction to Mathematical Reasoning22 chapters | 123 lessons | 13 flashcard sets

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we'll define binary relations. We'll look at examples in both a real-world context and a mathematical context to illustrate the concept of binary relations and to practice working with these types of relations.

Do you have a book that you keep your phone numbers in - a list of names of people you know along with their phone numbers? If so, then here's an interesting fact: that list is a binary relation! Wait, a what? That's a pretty fancy name. What the heck does it mean?

Technically speaking, in mathematics a **binary relation**, from a set *M* to a set *N*, is a set of ordered pairs, (*m*, *n*), or *m* and *n*, where *m* is from the set *M*, *n* is from the set *N*, and *m* is related to *n* by some rule. Confused yet? Let's see if we can put this into terms that we can better understand using your list of names and phone numbers.

If we let *M* be the set of all of the names of the people on your phone list, and we let *N* be the set of all of those phone numbers on the list, then your list relates each name in the set *M* to a number in the set *N*. Therefore, your list, which we'll call *L*, is a binary relation from the set *M* to the set *N*.

To clarify further, say that your friend Andy Smith has phone number 123-456-7891. Then the ordered pair (Andy, 123-456-7891) would be in the relation *L*, because Andy is in set *M* (the names), 123-456-7891 is in the set *N* (the phone numbers), and Andy is related to 123-456-7891 by the rule that 123-456-7891 is Andy's phone number.

Hmm. . . it's starting to make sense. Basically, binary relation is just a fancy name for a relationship between elements of two sets, and when an element from one of the sets is related to an element in the other set, we represent it using an ordered pair with those elements as its coordinates. Bingo! That's a binary relation!

That seems simple enough. Of course, these relations can be simple, as in our phone number example, or they can be more complicated. It all depends on the sets involved and the rule relating those sets. Let's explore this concept a bit further.

In our phone number example, we defined a binary relation, *L*, from a set *M* to a set *N*. We can also define binary relations from a set on itself. That is, we call a relation, *R*, from set *M* to set *M*, a binary relation on *M*.

For example, suppose you are at a work event with your coworkers, and a team building activity requires everyone at the event to pair up with someone that has the same hair color as them. If we let *Q* be the set of all of the people at the event, then this pairing off is a binary relation, call it *R*, on *Q*. Basically, *R* is the binary relation that consists of the ordered pairs (*q*1, *q*2), where *q*1 and *q*2 are elements of *Q*, and *q*1 has the same hair color as *q*2.

This is becoming more and more clear. Now that we are more familiar with the concept of binary relations, let's take a look at a binary relation in mathematics.

Consider the set *A* = {1,2,3,4,5,6,7,8,9}, and let â‰¥ be the relation on *A*, where (*x*,*y*) is in the relation â‰¥ if *x* is greater than or equal to *y*. This is an example of a binary relation from a set *A* to itself, so it's a binary relation on a set *A*. Now, let's see if we really understand this stuff. Let's consider the ordered pairs (5,2), (7,7), (3,9), and (10,8). Which of these ordered pairs would be in the relation â‰¥?

Well, let's think about it. The ordered pair (*x*,*y*) is only in the relation â‰¥ if both *x* and *y* are in the set *A*, and *x* is greater than or equal to *y*.

- First we'll look at (5,2). Both the numbers 5 and 2 are in the set
*A*, and 5 is greater than or equal to 2. Therefore, yes, (5,2) is in the relation â‰¥. - Next up is (7,7). The number 7 is in the set
*A*, and 7 = 7, so 7 is greater than or equal to 7. Therefore, the ordered pair (7,7) is also in the relation â‰¥. - The third ordered pair is (3,9). Both 3 and 9 are in the set
*A*, so that criteria is satisfied. However, notice that 3 is less than 9, so 3 is not greater than or equal to 9. Thus, the ordered pair (3,9) can't be in the relation â‰¥. - Lastly, we have the ordered pair (10,8). Recall that
*A*= {1,2,3,4,5,6,7,8,9}, so 10 is not in*A*. Even though 10 is greater than or equal to 8, 10 is not in*A*, so the ordered pair (10,8) can't be in the relation â‰¥.

Are you getting the hang of it? The more you work with binary relations, the more familiar they will become.

A **binary relation**, from a set *M* to a set *N*, is a set of ordered pairs, (*m*, *n*), where *m* is from the set *M*, *n* is from the set *N*, and *m* is related to *n* by some rule. We can also define binary relations from a set on itself. That is, we call a relation, *R*, from set *M* to set *M*, a binary relation on *M*.

These types of relations show up often in mathematics, and the concept can easily be extended to real life situations and scenarios. By being familiar with the concept of binary relations and working with these types of relations, we're better able to analyze both mathematical and real world problems involving them. Who knew that something as simple as a list of your acquaintances and their phone numbers could have such mathematical significance? You'll probably never look at that phone list the same again!

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
2 in chapter 9 of the course:

Back To Course

Math 97: Introduction to Mathematical Reasoning22 chapters | 123 lessons | 13 flashcard sets

- AFOQT Information Guide
- ACT Information Guide
- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- FTCE Middle Grades Math: Connecting Math Concepts
- Social Justice Goals in Social Work
- Developmental Abnormalities
- Overview of Human Growth & Development
- ACT Informational Resources
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- Cognition: Theory, Overview
- History of Sparta
- Realistic vs Optimistic Thinking
- How Language Reflects Culture & Affects Meaning
- Overview of Data Types in Java
- Managing Keys in Mobile Ad-Hoc Networks
- Using OpenStack for Building & Managing Clouds
- Quiz & Worksheet - Frontalis Muscle
- Octopus Diet: Quiz & Worksheet for Kids
- Logical Thinking & Reasoning Queries: Quiz & Worksheet for Kids
- Quiz & Worksheet - Fezziwig in A Christmas Carol
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- Special Education in Schools | History & Law
- Expert Advice on Bullying for Teachers | Bullying Prevention in Schools

- AP Environmental Science Textbook
- CBEST Test Prep: Practice & Study Guide
- Microbiology for Teachers: Professional Development
- Intro to Psychology Syllabus Resource & Lesson Plans
- MTTC Sociology (012): Practice & Study Guide
- ORELA General Science: Cell Structure & Function
- Dividing Fractions: Homework Help Resource
- Quiz & Worksheet - How to Teach Children Listening Skills
- Quiz & Worksheet - Visual Cliff Experiment
- Quiz & Worksheet - Christian Humanism in the Renaissance
- Quiz & Worksheet - Da Vinci's Mona Lisa
- Quiz & Worksheet - Meaning of Que Sera, Sera

- Monotheism in Christianity: Definition & Overview
- Plant Activities for First Grade
- Curriculum-Based Assessment Examples
- Inference Lesson Plan
- Common Core State Standards in Nevada
- Measurement Games for Kids
- Summer Tutoring Ideas
- New Mexico State Standards for Science
- Letter Writing Lesson Plan
- French and Indian War Lesson Plan
- US History Regents Essay Topics
- What is Professional Development for Teachers?

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject