We know we can use binary operations to add two numbers, x and y: Furthermore there are other operations such as or any other root and exponents. Operations can involve other mathematical objects other than numbers, such as sets.

Given two sets, A and B, we can define **multiplication** of these two sets as the **Cartesian product**. The new set is defined as

Before looking at abstract examples, consider this case:

e.g.1. Assume there is a student in a self-catering residence and they want to make food preps for the first four days in the week. They want to know how many possible

combinationsthey canmake using fruits (between grapes and apples) and meals (pasta and meatballs, chicken wrap).To solve this, let

Then the possible meal options are: (grapes, pasta and meatballs), (grapes, chicken wrap), (apples, pasta and meat balls) and (apples, chicken wrap).

The

Cartesian Productof sets A and B would be:

We can think of the above example in more abstract terms.

e.g.2 Let where = grapes, = apples, = pasta and meatballs, and finally = chicken wrap

So the Cartesian product

A more common example of the Cartesian product is the Cartesian plane. Two sets of Real numbers are multiplied together:

The representation of answers from the Cartesian products above is a list of ordered pairs

an **ordered pair** is a list of two things, x and y, enclosed in brackets and separated by a comma:

note: to understand this, we can think of the ordered pair as describing points on a plane. e.g.

e.g.1. is an ordered pair with thing 1 = (2, 4) and thing 2 = (4, 2)

e.g.2. is an ordered pair with thing 1 = library and thing 2 = police station

e.g.3. is an ordered pair with thing 1 = 2 and thing 2 = {2, 3}

Assume be two sets. Find set

To solve this, we will look at the product of the first element, 1, from set A with all the elements of B:

(Part 1)

Now, we will look at the product of the second element, 2, from set A with all the elements of B:

(Part 2)

Hence, for the Cartesian product, we put part 1 and part 2 together

an **ordered triplet** is a list of three things in brackets separated by commas:

We would represent this as

e.g.1. The Cartesian product of sets is

This is very different from

Hence produces an **ORDERED PAIR** while produces an **ORDERED TRIPLET**

In general,

So far, we’ve multiplied sets that are different but nothing is stopping us from multiplying two or more sets that are the same. For instances, we are familiar with the the Cartesian plane (xy-plane) which is defined as

can be simplified into This takes us to our next definition:

the Cartesian powers of a set A, where n = {1, 2, 3, ….}, is defined as

Hence,

e.g.1. is the set of all integers in 2D space

e.g.2. is the set of all integers in 3D space

First year maths lecture notes subject links – MathemafricaMay 9, 2018 at 9:02 pm[…] Cartesian products […]

GyashkaSeptember 24, 2018 at 12:48 pmWhy is the integer symbol also to the power of 2 in the example at the bottom of the page? Sorry if this is a silly question

YolisaSeptember 24, 2018 at 12:51 pmIt should be to the power of 3, thank you!