Notation for every odd integer number

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I have this equation:

$$f(x)=\tan(x)$$

I found the vertical asymptotes to be:

$$x=\frac{\pi}{2}k$$

What is the proper notation for that k is equal to every odd number integer(negative,positive, and zero)?

$$k\in\mathbb{Z}$$ is for every integer, but is there such a symbol for every odd number integer?

Natural numbers are positive, and sometimes zero counting numbers, my question is about integers not natural numbers.

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6 Answers

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they are $$(2k+1)\cdot \frac{\pi}{2}$$ with $$k \in \mathbb{Z}$$

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You can go with $2\mathbb Z +1$

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As long as we're considering alternatives, you could always write $$k\equiv 1\pmod 2$$

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Usually people write:

$$\frac{\pi}{2}(2k+1), k \in \mathbb{Z}$$

Sometimes people would use $\mathbb{O}$ for the set of all odd integers, but because it is not so standard they will tell you ahead of time:

$$\mathbb{O}=\{ 2n+1 : n \in \mathbb{Z}\}$$

So then, after defining $\mathbb{O}$, you would say:

$$\frac{\pi}{2}k, k \in \mathbb{O}$$

Get used the $\in$, it simply means "is a member of" some set.

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Alternatively, you could write

$$x = \frac{\pi}{2}k \quad , \quad k = \pm1, \pm3, \pm5 \dots$$

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You could do $x \epsilon \pi \mathbb{Z} / 2 \pi \mathbb{Z}$, without resorting to $k$.

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