Does there exist some notation to indicate that all elements of a set X are smaller than all elements of a set Y?
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4$\begingroup$ $\max X < \min Y$? $\endgroup$– angryavianCommented May 19, 2022 at 14:40
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2$\begingroup$ @angryavian that does not work if e.g. $X=(0,1)$ and $Y=\{1\}$. $\endgroup$– VeraCommented May 19, 2022 at 14:43
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$\begingroup$ @angryavian - Is there any guarantee that max (or sup) and min (or inf) exist? $\endgroup$– TaroccoesbroccoCommented May 19, 2022 at 14:47
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2 Answers
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You can state:$$\forall x\in X\forall y\in Y[x<y]$$I cannot find a shortcut by means of suprema, infima, maxima or minima. So if that's what you are looking for then this does not answer your question.
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$\begingroup$ Maybe also $y > \max X$ for all $y \in Y$? $\endgroup$– fennelCommented May 19, 2022 at 14:56
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1$\begingroup$ But there are situations where $X$ does not have a maximum. E.g. if $X=(0,1)$. A maximum of $X$ is by definition an element of $X$. $\endgroup$– VeraCommented May 19, 2022 at 14:58
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$\forall x \in X, \forall y \in Y, x < y$
