How to find the Cartesian Product

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.