There are more elements in the set $[-1,1]$ or in $[2,\infty]\cup [-\infty,-2]$

Question

Recently I have found this problem:

There are more elements in the set $[-1,1]$ or in $[2,\infty]\cup [-\infty,-2]$? Explain the reasons of your answer.

To be honest, I don't have completely any idea of how to approach this problem. I have found in Internet some videos (for example this https://www.youtube.com/watch?v=kf2Xwr22oGM), but I don't know if they can be useful to solve this problem. Any idea?

Hint: $x\mapsto x+\frac1x$ is a bijection between the sets $[-1,1]\setminus{0}$ and $(-\infty,-2]\cup[2,\infty)$. — Peter Foreman, Aug 30 '19 at 11:47
The concept of "having the same size" for infinite sets is cardinality. Some of the usual nice properties of the old concept do not apply anu more to infinite sets : a proper subset of an infinite sets may have the same cardinality of the "larger" one. — Mauro ALLEGRANZA, Aug 30 '19 at 11:48

score 1 · Answer 1 · answered Aug 30 '19 at 12:12

When talking about the "size" of the set or which set has "more elements" in this context we are talking about the cardinality of the sets.

Let $A$ and $B$ be sets. We have the following nice property:

$A\subseteq B \implies |A|\leq |B|$

Now... in the case that $A$ and $B$ both being finite, then $A\subsetneq B$ would have implied that $|A|\lneq |B|$, but that actually fails for infinite sets. It is possible to have a proper subset of a set have the same cardinality as the larger set in question. For example, the set of even integers is of the same cardinality as the set of integers.

We also have the nice anti-symmetry property that we expect, that if $|B|\leq |A|$ and $|A|\leq |B|$, then $|A|=|B|$, the cantor-schroder-bernstein theorem. Also, equality of cardinalities is transitive, again as we expect (proven by showing that composition of bijections is again a bijection).

Next, relevant to talking about the real numbers we have that any interval of real numbers has the same cardinality as the real numbers themselves.

There are many more nice properties about cardinality that I suggest you read up on, but these are the only things relevant here.

This all being said, you have $[-1,1]$ is a real interval and so has the cardinality of the real numbers. You also have that $(-\infty,-2]\cup [2,\infty)$ is a superset of an interval of real numbers and so has cardinality at least as great as that of the real numbers, while also being a subset of the real numbers and so has cardinality at most that of the real numbers and so has the cardinality of the real numbers.

It follows then that $[-1,1]$ and $(-\infty,-2]\cup [2,\infty)$ are both sets with cardinality equal to the real numbers and so have the same cardinality as eachother.

There are more elements in the set $[-1,1]$ or in $[2,\infty]\cup [-\infty,-2]$

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