Copyright
Like?

How to find the Cartesian Product

Start Your Free Trial To Continue Watching
As a member, you'll also get unlimited access to over 8,500 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.
Free 5-day trial
It only takes a minute. You can cancel at any time.
Already registered? Login here for access.
Start your free trial to take this quiz
As a premium member, you can take this quiz and also access over 8,500 fun and engaging lessons in math, English, science, history, and more. Get access today with a FREE trial!
Free 5-day trial
It only takes a minute to get started. You can cancel at any time.
Already registered? Login here for access.
  1. 0:05 A Cartesian Product Vacation
  2. 2:23 Example
  3. 3:41 Lesson Summary
Show Timeline
Taught by

Kathryn Maloney

Kathryn teaches college math. She holds a master's degree in Learning and Technology.

The Cartesian product allows us to take two sets of mathematical objects and create one new one. With one simple idea, the Cartesian product becomes quick and easy.

A Cartesian Product Vacation

Let's say we have two sets of free meals available on my vacation. Set A = {coffee, tea, milk}. These are my choices of free beverages. Set B = {breakfast, lunch}. These are the free meals with my vacation.

The Cartesian product, written A x B, is putting the elements from set A and elements in set B together. So the Cartesian product A x B = {(coffee, breakfast), (coffee, lunch), (tea, breakfast), (tea, lunch), (milk, breakfast), (milk, lunch)}. The Cartesian product is always written like an ordered pair: (first element, second element).

So I can have coffee with breakfast or lunch, tea with breakfast or lunch, or milk with breakfast or lunch. It is like you distribute set A into set B: coffee distributed to breakfast and lunch, tea distributed to breakfast and lunch, milk distributed to breakfast and lunch. It would give us our answer A x B = {(coffee, breakfast), (coffee, lunch), (tea, breakfast), (tea, lunch), (milk, breakfast), (milk, lunch)}.

If I have the Cartesian product B x A, we would have B x A = {(breakfast, coffee), (breakfast, tea), (breakfast, milk), (lunch, coffee), (lunch, tea), (lunch, milk)}. In this case, I can have breakfast with coffee, breakfast with tea, or breakfast with milk. I can also have lunch with coffee, lunch with tea, or lunch with milk. It is like you distribute set B into set A: breakfast distributed to coffee, tea, and milk; lunch distributed to coffee, tea, and milk. It would give us our answer B x A = {(breakfast, coffee), (breakfast, tea), (breakfast, milk), (lunch, coffee), (lunch, tea), (lunch, milk)}. This is the Cartesian product.

Example

Let's try another example. Here are a couple sets of clothes I brought on my vacation. Let set A = {shorts, skirt, pants}. Let set B = {shirt, swimsuit, sarong, blouse}. I have so many choices of what to wear; I don't know what to decide! Let's use the Cartesian product to show me all of the possible outfits.

I like to pick my bottoms first, so we will do A x B. Let's use distribute set A into set B: shorts distributed to shirt, swimsuit, sarong and blouse; skirt distributed to shirt, swimsuit, sarong and blouse; pants distributed to shirt, swimsuit, sarong and blouse. So the Cartesian product A x B = {(shorts, shirt), (shorts, swimsuit), (shorts, sarong), (shorts, blouse), (skirt, shirt), (skirt, swimsuit), (skirt, sarong), (skirt, blouse), (pants, shirt), (pants, swimsuit), (pants, sarong), (pants, blouse)}.

Wow! Now all I need to decide is which combination to wear! The Cartesian product is a great way to write out all of my options.

Lesson Summary

To find the Cartesian product, we can use idea of the distribution property. Take all of the elements in the first set multiplied with every element in the second set.

People are saying…

"This just saved me about $2,000 and 1 year of my life." — Student

"I learned in 20 minutes what it took 3 months to learn in class." — Student

See more testimonials

Did you like this?
Yes No

Thanks for your feedback!

What didn't you like?

What didn't you like?

Next Video
Create your Account

Sign up now for your account. Get unlimited access to 8,500 lessons in math, English, science, history, and more.

Meet Our Instructors

Meet all 53 of our instructors