Basic Cartesian prodcuts

Question

I am having some issues grasping basic ideas of Cartesian products.

I am reading a PDF my professor gave us explain sets/Cartesian products.

If $\mathbb{R}\times \mathbb{R}$ can be written as $\mathbb{R}^2$ then would $S^5$ be written $S \times S \times S \times S \times S$?

another question he asks us to do is describe the terms in the expression of $(x+1)^5$ and going with my line of thought on what $S^5$ is, I ended up doing $(x+1) (x+1) (x+1) (x+1)$ and multiplying them and taking that product putting it in a table with the remaining $(x+1)$ to get the elements of $(x+1)^5$

There is no textbook for this class and we had 2 snow days this week so I couldn't ask any questions in class (plus all the reading I'm doing online is making me more confused)

so if you guys could tell me if I am on the right path or show me where my "logic" is messed up I would greatly appreciate it.

Note: this is NOT homework, just questions asked to guide our understanding of notes.

Thanks!

Answer 1

It looks like you're getting two totally different ideas confused, which is understandable since the notation is very similar. The first notion is that of the arithmetic product that you're used to, and the second is that of the cartesian product.

The cartesian product is a neat way of denoting the set of all tuples consisting of elements from particular sets. Here's how the notation works: let's say $A = \{1,2,3\}$, $B = \{x,y\}$.

The set $A \times B$ would be the set of doubles $(a,b)$ where $a \in A$ and $b \in B$. So, $$ A\times B = \{(1,x),(2,x),(3,x),(1,y),(2,y),(3,y)\} $$ (confusingly, elements like these are occasionally written in the form $a \times b$ rather than $(a,b)$). Similarly, $A \times B \times \mathbb{R}$ would be the set of triples $(a,b,r)$ where $a \in A, b \in B, r \in \mathbb{R}$. You could even take a cartesian product of a set with itself. So, $B \times B$ would be $$ \{(a,a),(a,b),(b,a),(b,b)\} $$ With the usual shorthand, you can write this set as $B \times B = B^2$. Similarly, we'd have $$\overbrace{B \times \cdots \times B}^{n\text{ times}} = B^n $$ This notion is very much distinct from the arithmetic product. So, for example, $2 \cdot 3$ (which is $6$) means a very different thing than $\{2\}\times \{3\}$ (which is $\{(2,3)\}$) The first is the arithmetic product of numbers (or variables representing numbers), whereas the second is the cartesian product of sets. When you are calculating $(x+1)^5$, you are using the arithmetic product.

I hope that clears things up.

Answer 2

Instead of making things confusing and 'overly' theoretical, I would like you to imagine the Cartesian product of two sets in the following way:

$ \mathbb {R}^2 $ when interpreted as Cartesian coordinates is simply TWO axis (or number lines if you will) of real numbers. Imagine one horizontal and one vertical and you got a system that you have been working with since day 1. So when you multiply $ \mathbb{R} \cdot \mathbb{R} $ you get all the coordinates in that good old system. Now imagine $ \mathbb {Z}_+^2$ These are the coordinates of all integers above 0. So you'd start with (1,1), (1,2), (1,infinity), (2,1)...etc.

So what is $ \mathbb{R}^3 $? Slap another axis on there and you are done.

This way, abstract algebra and other subjects become much easier to comprehend rather than trying to come up with some theoretical definition for the sake of complexity.