## Icon Index Symbol • 7

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

Looking back over many previous discussions, I think one of the main things keeping people from being on the same page, or even being able to understand what others write on their individual pages, is the question of what makes a relation.

There’s a big difference between a single ordered tuple, say, $(a_1, a_2, \ldots, a_k),$ and a whole set of ordered tuples that it takes to make up a $k$-place relation.  The language we use to get a handle on the structure of relations goes like this:

Say the variable $x_1$ ranges over the set $X_1,$
and the variable $x_2$ ranges over the set $X_2,$ $\cdots$
and the variable $x_k$ ranges over the set $X_k.$

Then the set of all possible $k$-tuples $(x_1, x_2, \ldots, x_k)$ ranges over a set that is notated as $X_1 \times X_2 \times \ldots \times X_k$ and called the “cartesian product” of the “domains” $X_1$ to $X_k.$

There are two different ways in common use of defining a $k$-place relation.

1. Some define a relation $L$ on the domains $X_1$ to $X_k$ as a subset of the cartesian product $X_1 \times \ldots \times X_k,$ in symbols, $L \subseteq X_1 \times \ldots \times X_k.$
2. Others like to make the domains of the relation an explicit part of the definition, saying that a relation $L$ is a list of domains plus a subset of their cartesian product.

Sounds like a mess but it’s usually pretty easy to translate between the two conventions, so long as one watches out for the difference.

By way of a geometric image, the cartesian product $X_1 \times \ldots \times X_k$ may be viewed as a space in which many different relations reside, each one cutting a different figure in that space.

To be continued …

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