Select the correct answer below: $$ \begin{array}{l}\{n \mid 2 n+1 \in \mathbb{Z}\} \ \bigcirc\{2 n \mid n \in \mathbb{Z}\} \ \bigcirc\{2 n+1 \mid x \in \mathbb{Z}\} \ \bigcirc\{2 n+1 \mid n \in \mathbb{Z}\}\end{array} $$

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We wish to write the set of odd numbers in proper set‐builder notation. The standard way to write the set of odd integers is

$$
\{2n+1 \mid n \in \mathbb{Z}\}.
$$

Let’s analyze each option:

1. $$\{n \mid 2n+1 \in \mathbb{Z}\}$$

For any integer $$n,$$ the expression $$2n+1$$ is always an integer. Thus this set is actually

$$
\{n \in \mathbb{Z}\},
$$

which is just the set of all integers, not just the odd ones.

2. $$\{2n \mid n \in \mathbb{Z}\}$$

This describes the set of even integers because every even number can be written as $$2n$$ for some $$n \in \mathbb{Z}$$.

3. $$\{2n+1 \mid x \in \mathbb{Z}\}$$

In this expression the variable $$x$$ is used in the condition rather than $$n.$$ Since the variable in $$2n+1$$ is $$n,$$ writing $$x \in \mathbb{Z}$$ is not correct. (It is standard to use the same variable in both places.)

4. $$\{2n+1 \mid n \in \mathbb{Z}\}$$

This is the correct notation, since every odd integer can be written in the form $$2n+1,$$ with $$n$$ ranging over all integers.

Thus, the correct answer is

$$
\{2n+1 \mid n \in \mathbb{Z}\}.
$$
The correct answer is $$\{2n+1 \mid n \in \mathbb{Z}\}$$.

Select the correct answer below:

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