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Math 102: College Mathematics14 chapters | 108 lessons

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Lesson Transcript

Instructor:
*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.

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.

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 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.

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.

After watching this lesson, you should be able to interpret two mathematical sets to create a new set with the distributive property.

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3 in chapter 12 of the course:

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Math 102: College Mathematics14 chapters | 108 lessons

- Go to Logic

- Mathematical Sets: Elements, Intersections & Unions 3:02
- Cardinality & Types of Subsets (Infinite, Finite, Equal, Empty) 4:13
- How to Find the Cartesian Product 3:57
- Categorical Propositions: Subject, Predicate, Equivalent & Infinite Sets 4:24
- How to Change Categorical Propositions to Standard Form 3:28
- What is a Two-Way Table? 3:40
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