2

Similar to $x \ll y$ or $x \lll y$, I'm wondering if there is a corresponding symbol for subsets.

For example:

$$x \subset \subset y$$

Something to represent $x$ is subset of $y$, but much smaller in size than $y$.

I realize one could simply say $x \subset y \wedge |x| \ll |y|$, but this seems a bit verbose and I'm also genuinely curious if this symbol exists.

ryan
  • 81
  • 2
    The symbol $\ll$ is not really used in math at all, and is mostly used exclusively in the physical sciences. This is due to the fact that math deals in arbitrary degrees of accuracy, and such a symbol would not be (usefully) defined. – George Jul 09 '17 at 23:54
  • One should remember that $\ll$ is used informally in sciences to express a diference of several orders of magnitude (for example decimal places in measurements of millions). Maybe you can try to make an analogy betwen orders of magnitude of numbers, and orders of "size" in sets. – omega-stable Jul 09 '17 at 23:54
  • @George would you recommend that I move this to another stack exchange? Regardless of whether it's used a lot or not, I'm still curious if it exists. There are problems I have come across where I believe it would be useful for emphasizing the size of a subset. – ryan Jul 09 '17 at 23:57
  • As @George said, decimal places next to millions may mean little in science, but in mathematics $1 000 000 \neq 1 000 000.1 $ strictly. On the other side, an error of 10^6 can be small when dealing with things in number theory – omega-stable Jul 09 '17 at 23:57
  • In factt, $\ll$ is used in measure theory for "absolutely continuous". – GEdgar Jul 10 '17 at 00:23
  • Symbol $A \subset\subset B$ is also used in math, but it means the closure of $A$ is contained in the interior of $B$ or something like that. – GEdgar Jul 10 '17 at 00:24
  • https://math.stackexchange.com/questions/1516976/much-less-than-what-does-that-mean might be a good read, also I can think of $x\in {\neg z: |z|\approx |y|}$, but again that's long. –  Jul 10 '17 at 00:26
  • @RoddyMacPhee, I have read that. It had actually inspired the question, wondering if $\ll$ translated to sets. – ryan Jul 10 '17 at 00:28

2 Answers2

1

As far as I know, there is no commonly adopted symbol for the scenario you describe. It is always best practice to use words where symbols could suffice for maximum legibility. There are exceptions to this rule, of course, and if you want to introduce notation, you should feel free to do so only after you've defined it for your readers.

In your example, if you want to make a symbol to mean "a set has cardinality much less than another," first take care to define precisely what you mean by that, and then pick your favorite symbol for the shorthand, being sure to spell it out for your readers.

For example, let's define what it means for a finite set $A$ to have cardinality much less than that of $B$ and invent our own notation for it:

Definition: Let $A$ and $B$ be two finite sets. If $|A| < |B|-5$ then we say the cardinality of $A$ is much less than the cardinality of $B$, and in this case we write $A\preccurlyeq B$.

Now whenever we want to use this shorthand, we can do so in the confidence that our readers know exactly what we mean.

Edit: If you are just trying to impart some intuition to your readers, it would probably be fine to write something like $|A|\ll|B|$ to convey the idea that the cardinality of $A$ is much less than the cardinality of $B$. But even then, it would be better to just say "$A$ has cardinality much less than that of $B$" rather than risk your readers misinterpreting the intuition you are trying to give them because they don't know how to read a symbol that doesn't exist for this exact purpose.

Alex Ortiz
  • 26,211
  • 1
    I like this, but I think the purpose of "much less than" is that it is imprecise. So would there be a way to define it that captures this impreciseness? I guess one could arbitrarily define it in the context of subsets based on $\ll$. – ryan Jul 10 '17 at 02:08
  • @Ryan, even in scenarios where you don't have exact information, you still need to be precise about what you mean. If you have heard of big-O notation, we throw away some information, but we are still exactly sure about what we mean when we say that $f(x) = O(g(x))$ as $x\to\infty$, for example. You cannot throw away this precision because it is imperative that we know exactly what we are talking about in math and applications, even when we don't have exact information, or we don't care about some of the information. It depends on the context and your purposes how you choose to define it. – Alex Ortiz Jul 10 '17 at 02:12
  • Yes, big-O notation is precise and explicit in its definition. What I was aiming for was something similar to $\ll$, where it is informal, imprecise. It should convey a more human perception of the inequality rather than a formal mathematical one. $\ll$ is clearly more for the human reader. To base a proof on $\ll$ but not on $<$ would be non-sensical without a more rigorous definition. – ryan Jul 10 '17 at 02:21
  • 1
    @ryan If that is what you are looking for, then of course you recognize it has no real place when it comes to doing formal mathematics, but it should be fine to write something like $|A|\ll|B|$ to convey the idea that the cardinality of $A$ is much less than the cardinality of $B$ if you are just trying to impart some intuition to your readers. But even then, it would be better to just say "$A$ is much smaller than $B$" rather than risk your readers misinterpreting the intuition you are trying to give them because they don't know how to read a symbol that doesn't exist for this exact purpose. – Alex Ortiz Jul 10 '17 at 02:23
  • Thank you, yes that is what I'm wondering. If there is a pre-existing symbol to represent this notion rather than the merit of such a symbol or even merit of the concept. It is mostly to satisfy my curiosity rather than satisfy a formal problem. – ryan Jul 10 '17 at 02:28
  • 1
    @ryan Right. I think the closest match to what you are looking for is $\ll$, but I am confident that such a symbol has not been formulated for this exact purpose before. – Alex Ortiz Jul 10 '17 at 02:30
0

The notion of size in a set is given by the cardinality in a set. As I commented, mathematics deals with arbitrary degrees of precision. For finite sets, a simple counting argument could show which set is larger, ie: which one has larger cardinality, but this type of argument fails for infinite sets. One has to characterize "how many" elements there are in a set using a different concept, and that concept is functions.

To illustrate the idea of how it's done, imagine a set $A$ of $7$ elements and another set $B$ of $5$. Any function $A\to B$ will necessarily be not injective; you can start by assigning the first $5$ elements to distinct elements in $B$, but the last $2$ will always be mapped non-injectively. Similarly, you can't find a function $B\to A$ that is surjective.

Cardinality is then extended to infinite sets by the definition that two sets have the same cardinality if there exists a bijection between them. Unintuitively, we find that the integers have the same cardinality as the rationals, and again, that the rationals don't have the same cardinality as the real numbers. We define the lowest infinity to be that of the integers, and call a set countably infinite set if there exists a bijection with the integers.

We can therefore quantify how large a set sits within another set by having some sort of measure on their cardinality.

A term that is used in measure theory that is the "equivalent" to $\ll$ is the rigorous term "almost all". The definition for almost all is "all but countably many". For example, within the reals, almost all elements are irrational numbers, since there are only countably many rationals.

George
  • 2,656
  • I appreciate the content and insight, but I'm unclear on how this answers my question. I understand your rational for the cardinality of sets, and how to compare infinite sets, but I do not see any claim about notation for this concept. – ryan Jul 10 '17 at 00:37
  • In the last paragraph, I mention the term used, which is 'almost all', or conversely, 'almost everywhere', which is defined to be all except countably many cases. Alternatively, a set could have Lebesgue measure $0$ (which is defined analogously) – George Jul 10 '17 at 00:43
  • 1
    I doubt such a symbol (if that's what you're looking for) has been defined, given that defining it would probably waste more time/space than the amount of times it would be used! – George Jul 10 '17 at 00:49
  • So I would say $x$ is "almost all" of $y$? or would it be $x$ is not almost all of $y$ for $|x| \ll |y|$, and would this be the same as $x$ is almost none of $y$ ? – ryan Jul 10 '17 at 00:50
  • You make a good point, but I don't think I would call it a waste. If it can more clearly and concisely get the point across than a heap of words for at least one use than it would not be a waste. – ryan Jul 10 '17 at 00:53
  • "Almost all" is not exactly "all but countably many." When you describe a property $P$ that holds for "almost all" $x$ in a set $S$, you mean there is a subset $Z$ contained in $S$ of measure $0$ such that $P(x)$ holds for all $x$ in $S\setminus Z$. This includes "all but countably many" as a special case in the context of the Lebesgue measure. – Alex Ortiz Jul 10 '17 at 01:50
  • For instance, there are uncountable sets of Lebesgue measure $0$, like the Cantor middle-thirds set. – Alex Ortiz Jul 10 '17 at 02:08