The discussion centers on the kernel of the cosine function, specifically denoted as ##Ker[\cos(x)]##, which is expressed as either ##\pi \mathbb{Z} - \frac{\pi}{2}## or ##\pi \mathbb{Z} + \frac{\pi}{2}##. Both notations represent the same set of values, which can be succinctly written as ##\{(2n+1)\frac{\pi}{2} \mid n \in \mathbb{Z}\}##. The conversation also suggests introducing a notation for odd integers, ##\mathbb{Z}_{\text{odd}}##, to simplify expressions involving the kernel of the cosine function. Additionally, LaTeX formatting tips for the kernel and cosine function are provided.
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Fredrik said:Assuming that the former notation means
$$\left\{\pi n-\frac\pi 2\,\big|\,n\in\mathbb Z\right\},$$ and the latter means the same thing with + instead of -, then both notations represent the same set. I think I would just write
$$\left\{(2n+1)\!\frac\pi 2\,\big|\,n\in\mathbb Z\right\},$$ because "simple" notations like yours tend to require explanation. If you want a simple notation, then why not introduce a notation for the set of odd integers, say ##\mathbb Z_\text{odd}## and write
$$\mathbb Z_\text{odd}\frac\pi 2?$$
LaTeX tips: \operatorname{ker} and \cos x. ##\operatorname{ker}[\cos x]##