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I want to show these duplets in a compact form $$(\pm x,+y)\quad,(\pm x,-y)$$ Then, is it clear to use $$(\pm x,\pm y)$$ or this only considers upper and lower signs together?

charmin
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    A good question! Does $(\pm x,\pm y)$ denote two points, or four? For me, it is not at all clear. – TonyK Apr 05 '21 at 12:27
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    @charmin $|x|=\pm x,|y|=\pm y\ (\pm x,\pm y)\to (|x|,|y|)$ – Khosrotash Apr 05 '21 at 12:28
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    sometimes, clarity is much better than compactness. – peek-a-boo Apr 05 '21 at 12:30
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    @Khosrotash $|x|$... a single value... is not equivalent to $\pm x$ which denotes two values. The set ${(\pm 1, 2)}$ is commonly interpreted as being the set ${(1,2),(-1,2)}$ while the set ${(|1|,2)}$ is interpreted only as the the set ${(1,2)}$ – JMoravitz Apr 05 '21 at 12:32
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    If you insist on writing this, perhaps write it as ${(c_1 x, c_2 y)~:~c_i\in{-1,1}}$ which extends easily to $n$-tuples. – JMoravitz Apr 05 '21 at 12:33
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    Related: https://math.stackexchange.com/q/4045621/42969 – Martin R Apr 05 '21 at 12:50
  • @JMoravitz: "If you insist on writing this": the OP doesn't insist on anything, they just want to know the best way to denote the set ${{x,y},{x,-y},{-x,y},{-x,-y}}$. – TonyK Apr 06 '21 at 15:17

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I prefer $\pm(\pm x, y)$, but do not see this idiom widely-used. To the contrary, students sometimes ask if it means the same as $(\pm x, \pm y)$, whereupon we have to have the Discussion about Which Choices of Sign are Intended.

Not that you asked, but $\pm(x, y)$ is (to my eye) preferable to denote the two points with same sign.

Andrew D. Hwang
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