1:55 PM
You can also regard C
to be a relation between some proper subsets of ℝ
. There are no x
and y
for which xCy
and |x| > 1
or |y| > 1
, for instance. So you could regard C
as a relation between [-1, 1]
and itself. (As you probably know, [-1, 1]
means the closed interval from -1
to 1
, so in this context, i.e., over the reals, [-1, 1] ⊆ ℝ
and, specifically, [-1, 1] = {x ∈ ℝ | -1 ≤ x ≤ 1}
.) It is true that C ⊆ [-1, 1] × [-1, 1]
.
I don't know in the UK, but in the US there is an incessant demand, at least up through high school, that math students draw the arrows on their axes to indicate they go on forever. A student might defend their failure bold choice not do so, on a graph of x² + y² = 1
, by saying they were viewing the relation not as between ℝ
and ℝ
but rather between the bounded subsets of ℝ
indicated by the axes that were drawn.
One might imagine another student who draws their x
axis as a line (with arrows on the ends, lest it be a line segment) together with some pictures of goats (the elements of G
) and their y
axis as a line (with arrows on the ends, lest it be a line segment) together with some pictures of doorknobs (the elements of D
).
These goat and doorknob pictures wouldn't be separate from the axes, but rather should be thought of as part of the axes, since this hypothetical students is thinking of how C ⊆ (ℝ ∪ G) × (ℝ ∪ D)
.
In all seriousness, though, whilst I am unconvinced of the wisdom of deducting points for the failure to draw arrows on axes, the way one draws axes does potentially indicate what kinds of things one is claiming that one's plot provides information about.
*this hypothetical student is thinking
Consider a binary relation F
where xFy
means "x
has y
as a friend." If this relation is symmetric then we could just say "x
and y
are friends." Let's go with that.
I'll call Alice, Bob, Cassidy, Derek, and Eve by the short names a
, b
, c
, d
, and e
, respectively. Also, by "Eve," I actually mean the abstract idea of of an office photocopier that is able to output at least three pages per second and has built-in stapling capabilities. Also, I am not undertaking any trickery here; none of Alice, Bob, Cassidy, or Derek is Eve. Not only that, no two of those names are names of the same person (i.e., Alice, Bob, Cassidy, and Derek are four people).
Suppose Alice is friends with both Bob and Cassidy but, regrettably, Derek has no friends. Then we know:
(a, b) ∈ F
(a, c) ∈ F
(b, a) ∈ F
(c, a) ∈ F
∀x (d, x) ∉ F
∀x (x, d) ∉ F
With usual semantics and assumptions about abstract ideas of office equipment, it is also true that:
∀x (e, x) ∉ F
∀x (x, e) ∉ F
However, it does not seem like the e
-sentences, such as "∀x (e, x) ∉ F
", are relate to the same kind of facts as the d
-sentences, such as "∀x (d, x) ∉ F
".
This is not to say that we wish to deny either of them, or even to define F
in such a way as to falsify any of them...
...but rather than it might sometimes be of interest to define F
in such a way as to contain enough information to distinguish between that which is friendless because it's unclear what it would mean for it to have a friend and F
is not intended to cover it, and that which is friendless due to the unfortunate circumstance of not having any friends, which is the sort of thing F
is intended to cover.
Note that I have not written an expression for F
in terms of (some of) a
, b
, c
, d
, and e
, because there is not enough information to do so. For example, are Bob and Cassidy friends? I have not given enough information to tell. However, I might want to express what I know about friendship. Suppose G
is the relation that holds between x
and y
when I know x
to be friends with y
. That is, "xGy
" expresses "Eliah knows x
has y
as a friend."
Suppose further that my knowledge, while it may be incomplete, is not erroneous. Note that G ⊆ F
, and, taking the above information as the extent of my knowledge on the matter:
G = {(a, b), (a, c), (b, a), (c, a)}
We of course have ∀x (d, x) ∉ G
. Now consider Florence, a person other than Alice, Bob, Cassidy, and Derek. Take "f
" to mean "Florence". Then ∀x (f, x) ∉ G
too. However, it may well be that Florence has friends, even among Alice, Bob, and Cassidy. (But not Derek, unfortunately.)
It might be handy for the relation G
to be able to distinguish between d
, which G
does not pair with anything due to friendlessness, and f
, which G
does not pair with anything because it is outside the scope of G
's purpose to do so since G
is about my (Eliah's) knowledge. But G
cannot do so, because it is constructed as its graph.
More formally, what I mean is that I cannot express, in terms of G
, d
, and f
alone, the way d
and f
differ under G
... because in fact they do not differ under G
at all, but instead under further information we carry around with us as we think and write about G
.