Are there more real numbers in the interval $[1,\infty)$ than in the interval $(0,1]$? Or not?

Question

We all might be familiar with the beautiful method Cantor devised to prove that the cardinality of the set of real numbers is more than that of the set of natural numbers (Refer to: https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument). His method involved mapping each natural number to unique real number, no two natural numbers can map to same real number, and then showing that there are always real numbers left which are not assigned to any natural number.

But what can we say about the cardinalities of the sets $A$ and $B$ defined as $$A=\{x \ |\ x\in (0,1]\}$$ $$B=\{x \ |\ x\in [1,\infty)\}$$

Is the cardinality of $A,$ written as $|A|$ less than or equal to $|B|$?

I think that $|A|=|B|$ and I think so based on the following reasoning. Now I would use somewhat similar argument as Cantor did, in the sense that I will define one to one mapping of elements from set $A$ to set $B$.

Let us define a function, $f:[1,\infty)\rightarrow (0,1]$ defined by $$f(x)=\frac{1}{x}$$ It's easy to see that this is a one-one function. So we have mapped every real in $A$ to some unique real in $B$. And we have done this for all the elements of $A$ and $B$, as is clear from the fact that this is an invertible function.

So what's wrong in saying that this argument proves $|A|=|B|$?

Please express your thoughts on this.

Thanks:)

Answer 1

So what's wrong in saying that this argument proves $|A|=|B|$?

Absolutely nothing. If a one-to-one correspondence exists between two sets, then they have the same cardinality.

Infinite sets are just weird. For example, all of the following sets, even though intuitively having different sizes, have the same cardinality (denoted $\aleph_0$):

• The integers $\mathbb{Z}$.
• Subsets of $\mathbb{Z}$
  • The set of natural numbers $\mathbb{N}$.
  • The set of prime numbers.
  • The set of odd numbers.
  • The set of powers of 2.
  • The set of numbers that have a Collatz sequence that reaches 1, whether the conjecture turns out to be true or false.
  • The set of integers that, when written in binary and interpreted as a UTF-8 string, contain the name of a character from a Shakespeare play.
• Supersets of $\mathbb{Z}$:
  • The set of rational numbers $\mathbb{Q}$.
  • The set of Gaussian integers.
  • The set of constructible numbers.
  • The set of computable numbers, under any computation model in which programs and their possible inputs and outputs consist of a finite but unbounded sequence of symbols from a finite alphabet.

Question: Can you think of a set of numbers that has more elements than $\mathbb{Z}$, but fewer than $\mathbb{R}$?