We know we can use binary operations to add two numbers, x and y: $x+y, x-y, x \times y, x \div y.$ Furthermore there are other operations such as $\sqrt{x}$ or any other root and exponents. Operations can involve other mathematical objects other than numbers, such as sets.

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

$A \times B = \{(a,b): a \in A, b \in B\}$

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 combinations they can make using fruits (between grapes and apples) and meals (pasta and meatballs, chicken wrap).

To solve this, let  $A = \{ \text{ grapes, apples } \} \text{ and } B = \{ \text{pasta and meatballs, chicken wrap} \}$

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

The Cartesian Product of sets A and B would be:

$\text{A x B} = \{( \text{ grapes, pasta and meatballs}), (\text{ grapes, chicken wrap }), (\text{ apples, pasta and meat balls }), (\text{ apples, chicken wrap}) \}$

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

e.g.2 Let $\text{A} = \{a_1, a_2\} \text{ and } \text{B} = \{b_1, b_2\}$ where $a_1$ = grapes, $a_2$ = apples, $b_1$ = pasta and meatballs, and finally $b_2$ = chicken wrap

So the Cartesian product $\text{A x B} = \{(a_1, b_1), (a_1, b_2), (a_2, b_1), (a_2, b_2)\}$

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

$\mathbb{R} \times \mathbb{R} = \{(x,y): x, y \in \mathbb{R} \}$

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

$def^n$ an ordered pair is a list of two things, x and y, enclosed in brackets and separated by a comma: $(x, y)$

note: $(x,y) \ne (y, x) \text{ unless } x = y$ to understand this, we can think of the ordered pair as describing points on a plane. e.g. $(2, 4) \ne (4, 2)$

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

e.g.2. $(\text{ library }, \text{ police station })$ is an ordered pair with thing 1 = library and thing 2 = police station

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

Assume $A = \{p, q\} \text{ and } B = \{\frac{1}{2}, 3, {0,1}\}$ be two sets. Find set $A \times B$

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) $\{ (p, \frac{1}{2}), (p, 3), (p, 0), (p, 1) \}$

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

(Part 2) $\{ (q, \frac{1}{2}), (q, 3), (q, 0), (q, 1) \}$

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

$A \times B = \{ (p, \frac{1}{2}), (p, 3), (p, 0), (p, 1), (q, \frac{1}{2}), (q, 3), (q, 0), (q, 1) \}$

$def^n$ an ordered triplet is a list of three things in brackets separated by commas: $(x,y,z)$

We would represent this as $A_1 \times A_2 \times A_3=\{ (x_1, x_2, x_3): x_1 \in A_1, x_2 \in A_2, x_3 \in A_3 \}$

e.g.1. The Cartesian product of sets $\mathbb{R}, \mathbb{N} \text{ and } \mathbb{Z}$ is $\mathbb{R} \times \mathbb{N} \times \mathbb{Z} = \{(x, y, z): x \in \mathbb{R}, y \in \mathbb{N} , z \in \mathbb{Z}\}$

This is very different from $\mathbb{R} \times (\mathbb{N} \times \mathbb{Z}) = \{ (x, (y, z) ): x \in \mathbb{R}, (y, z) \in (\mathbb{N} \times \mathbb{Z} ) \}$

Hence  $\mathbb{R} \times (\mathbb{N} \times \mathbb{Z})$ produces an ORDERED PAIR while $\mathbb{R} \times \mathbb{N} \times \mathbb{Z}$ produces an ORDERED TRIPLET

In general,

$A_1 \times A_2 \times .... \times A_{n-1} \times A_n = \{(x_1, x_2, ...., x_n): x_i \in A_i \text{ for } i = 1, 2, ..., n\}$

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

$\mathbb{R} \times \mathbb{R} = \{ (x, y): x, y \in \mathbb{R}\}$

$\mathbb{R} \times \mathbb{R}$ can be simplified into $\mathbb{R}^2.$ This takes us to our next definition:

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

$A^n = A \times A \times .... \times A \text{ ( n-times )}$

Hence, $A^n = \{(x_1, x_2, ..., x_n):x_1, x_2, .., x_n \in A \}$

e.g.1. $\mathbb{Z}^2 = \{(m, n): m, n \in \mathbb{Z} \}$ is the set of all integers in 2D space

e.g.2. $\mathbb{Z}^3 = \{(m, n, p): m,n, p \in \mathbb{Z} \}$ is the set of all  integers in 3D space

 How clear is this post?