# plotting plot $(- \infty, x]$ for any arbitrary $x$? [closed]

is it possible to plot $$(- \infty, x]$$ for any arbitrary $$x$$ on wolfram online? is it possible provide commands?

I am trying to understand the concept of Vapnik-Chervonenkis dimension.

• No. No finite system can handle plotting an infinite quantity. You will have to provide more details to even try to answer your question. – MarcoB Dec 6 '20 at 20:25
• I plotted over $[0, \infty)^2$ here: mathematica.stackexchange.com/a/222911/4999 -- is that what you're after? – Michael E2 Dec 6 '20 at 20:35
• I don't see how this question is specific to wolfram-cloud. The solution should be the same on any platform. (That's why I removed the tag.) – Michael E2 Dec 6 '20 at 20:38
• It would be more helpful if you included a concrete example of what you wanted to plot. – Michael E2 Dec 6 '20 at 21:05
• would you please provide both of them as an answer? @MichaelE2 – M K Dec 6 '20 at 22:27

## 3 Answers

From https://www.cs.cornell.edu/courses/cs683/2008sp/lecture%20notes/683notes_0428.pdf, which is linked on the MathWorld link in a comment by the OP, I inferred that what was desired was the illustration of the following concept. I will us the interval $$(-\infty, r]$$ instead of the OP's notation so as not to conflict with the notation in the definitions:

A set system $$(x, S)$$ consists of a set $$x$$ in the definition along with a collection of subsets of $$x$$. A subset...$$A ⊆ x$$ is shattered by $$S$$ if each subset of $$A$$ can be expressed as the intersection of $$A$$ with a subset in $$S$$.

VC-dimension of a set system is the cardinality of the largest subset of $$A$$ that can be shattered.

1D: There isn't much detail in the OP. If the set $$x$$ is the set of real numbers and $$S$$ is the collection of intervals $$(-\infty, r]$$, then the intervals may be represented by HalfLine.

Update: Here's a way to visualize the concept with NumberLinePlot and "plot $$(-\infty,r]$$":

SeedRandom[0];
Block[{r, s, A},
A = RandomReal[{-2, 2}, 3];
r = Mean[A];
s = Interval[{-Infinity, r}];
NumberLinePlot[{s, A}, PlotLegends -> {"s", "A"}]
]


A 3-element $$A$$ cannot be shattered since the only subsets that may be obtained are the empty set, $$\{\,x_1\,\}$$, $$\{\,x_1,x_2\,\}$$, and $$\{\,x_1,x_2,x_3\,\}$$. (The VC dimension is $$1$$, I believe.)

2D: If $$x$$ is the real plane and $$S$$ consists of half-planes $$\{\,(a,b) \mathrel{|} a < r\,\}$$, then InfiniteLine and HalfPlane may be appropriate.

There are several ways you can plot a function over an infinite domain, by coming up with a function that compresses that domain to a finite interval.

For example, plotting Sin[x] in the interval (−∞,5] by scaling the domain with x |-> 5*2 x/(x + 1)

(-1 maps to −∞, 0 maps to 0, 1 maps to 5)

f[x_] := Sin[x]

Plot[f[5* 2 x/(x + 1)], {x, -1, 1}, Ticks -> {{{-1, "-\[Infinity]"}, {1, "5"}}, Automatic}]


Maybe this

x=5;
Plot[Sin[t], t ∈ ImplicitRegion[-∞ < t <= x, t]]

Manipulate[
Plot[Sin[t], t ∈ ImplicitRegion[-∞ < t <= x, t],
PerformanceGoal -> "Quality"], {{x, 5}, -50, 10}]