Projection Functions: Combining X1 to Xn Coordinates

[Square1]

Given sets X1,...,Xn the projection from the product X1 * ... *Xn is the function
pri = prxi : X1 * ... *Xn → Xi , pri(t1, ..., ti, ...,tn) := ti

Im having a hard time figuring out what exactly is going on here.

Some coordinate from all sets X1 to Xn are getting multiplied together? The above bit is an introductory example/case of "the projection function". Any help will do. Thanks.

 

[jgens]

Let $X_1,\dots,X_n$ be sets. Then $X_1 \times \cdots \times X_n = \{(x_1,\dots,x_n) : x_i \in X_i\}$; that is, elements of $X_1 \times \cdots \times X_n$ are $n$-tuples where the $i$-coordinate is contained in $X_i$.

Now define the projection mapping $\pi_i:X_1 \times \cdots \times X_n \rightarrow X_i$ by setting $\pi_i(x_1,\dots,x_i,\dots,x_n) = x_i$. So $\pi_i$ finds the $i$-th value in the $n$-tuple $(x_1,\dots,x_n)$ and then prints out this value.

 

[Square1]

So..

You are multiplying any number of sets (any restrictions on "n" of the sets?), and the resulting (insert word here...set??) is a collection of all possible n-tuples (x1,...xn) where any given xi of a tuple will be a member of set Xi (so does this imply a condition on the order of multiplying the sets?).

Is that right?
The next line though I am a bit confused about the choice of pi. :(

 

[jgens]

You are multiplying any number of sets (any restrictions on "n" of the sets?), and the resulting (insert word here...set??) is a collection of all possible n-tuples (x1,...xn) where any given xi of a tuple will be a member of set Xi (so does this imply a condition on the order of multiplying the sets?).

There is no restriction on the number of sets. You can actually generalize this to take infinite products of sets, although at that point you have to abandon the tuple notation.

Strictly speaking the order in which you take products of the sets is important. For example, $\{1\} \times \{0\} \neq \{0\} \times \{1\}$. Under certain circumstances the order is more or less irrelevant, but since you seem to be just learning about cartesian products, I would not worry about it.

The next line though I am a bit confused about the choice of pi. :(

Sorry about that. A common name for the $i$-th projection function is $\pi_i$. This has no connection with the number we denote by $\pi$ or anything else. This is kind of an example where symbols get overloaded in math.

 

[Square1]

Ah ok thank you thank you.

Now define the projection mapping $\pi_i:X_1 \times \cdots \times X_n \rightarrow X_i$ by setting $\pi_1(x_1,\dots,x_i,\dots,x_n) = x_i$. So $\pi_i$ finds the $i$-th value in the $n$-tuple $(x_1,\dots,x_n)$ and then prints out this value.

So picking i=1 for example means, you will be "extracting" a single n-tuple then from the collection X1 x ... x Xn , where all entries are are the i'th/first coordinate of their original set?

 

[jgens]

So picking i=1 for example means, you will be "extracting" a single n-tuple then from the collection X1 x ... x Xn , where all entries are are the i'th/first coordinate of their original set?

Not really. The projection functions do not extract an $n$-tuple. Rather they extract a particular value from each $n$-tuple you give them as an input. An example might be helpful.

Consider the case when $X_1 = \cdots = X_n = \mathbb{R}$. Then $X_1 \times \cdots \times X_n = \mathbb{R}^n$. We will consider the first projection function $\pi_1$. Notice that $(1,0,\dots,0) \in \mathbb{R}^n$ and $(1,2,3,4,\dots,n) \in \mathbb{R}^n$. It turns out that $\pi_1(1,0,\dots,0) = 1$ and that $\pi_1(1,2,3,4,\dots,n) = 1$.

So given an $n$-tuple input, the projection function searched for the $i$-th coordinate and then printed out whatever value was on that coordinate.

 

[Square1]

Hmm am i getting closer? So are projection functions can be used to output both: a given coordinate of an n-tuple, or a given i'th set, depending on how you define "pi sub i"
ex: pi*sub*Xi : X1 x ... x Xn
or
pi*sub*i : (t1, ... ti, ...tn)

? (crossed fingers)

 

[jgens]

So are projection functions can be used to output both: a given coordinate of an n-tuple, or a given i'th set, depending on how you define "pi sub i"

Not really. I mean $\pi_i$ takes an $n$-tuple in $X_1 \times \cdots \times X_n$ as an input, and then outputs the $i$-th coordinate of that $n$-tuple.

It is true that $\pi_i(X_1 \times \cdots \times X_n) = X_i$ and you can define a set-function which will output the image of a given set under $\pi_i$, but these are not the functions one usually refers to when talking about projection functions.

 

[Square1]

So which n-tuple does it take from X1 x ... x Xn ?

 

[jgens]

So which n-tuple does it take from X1 x ... x Xn ?

You can apply $\pi_i$ to any $n$-tuple in $X_1 \times \cdots \times X_n$.

 

[Square1]

Ok wait a second, when you say here the elements of the resulting set multiplication are...

Let $X_1,\dots,X_n$ be sets. Then $X_1 \times \cdots \times X_n = \{(x_1,\dots,x_n) : x_i \in X_i\}$; that is, elements of $X_1 \times \cdots \times X_n$ are $n$-tuples where the $i$-coordinate is contained in $X_i$.

{(x1, ..., xn)}, is the tuple inside this set here just one element of the resulting set, implying many more such elements, or are the parenthesis not indicating one element but the entire set?

 

[jgens]

{(x1, ..., xn)}, is the tuple inside this set here just one element of the resulting set, implying many more such elements, or are the parenthesis not indicating one element but the entire set?

In general, there are many elements in $X_1 \times \cdots \times X_n$. How many elements exactly depends on the sizes of $X_1,\dots,X_n$. It might help if you read the article here: http://en.wikipedia.org/wiki/Cartesian_product

 

[Square1]

Ok I think was
a) overthinking this too much lol, and
b) overlooked two points (the last question I asked, and that by definition the projection function outputs a single number...at least in my class notes.)

I think I got it now. Thanks for the help, and patience :D