Question

Check whether the set of integers is countable or not.

Answer 1

Answer on Question #79361 – Math – Real Analysis

Question

Check whether the set of integers is countable or not.

Solution

The set of natural integers is countable. Let $N$ denote the set of integers,

$Z = \{0, 1, 2, 3, \ldots\}$ denote the set of natural numbers together with .

To prove that the set $N$ of integers is countable it is sufficient to construct a bijection

$$f: N \to Z.$$

We construct it in such a way:

$$f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2, f(4) = -2, f(5) = 3, \dots$$

The formula for this function is

$$f(n) = (-1)^{n+1} \left| \frac{n}{2} \right|,$$

where $[\cdot]$ denotes the ceiling function.

The inverse map is

$$f^{-1}(k) = 2|k| - 1_{(0, +\infty)}(k).$$

That’s why this function is really a bijection.

Answer: the set of integers is countable.

Answer provided by https://www.AssignmentExpert.com