# Notation: set X whose elements are smaller than each elements of set Y

Does there exist some notation to indicate that all elements of a set X are smaller than all elements of a set Y?

• $\max X < \min Y$? May 19 at 14:40
• @angryavian that does not work if e.g. $X=(0,1)$ and $Y=\{1\}$.
– Vera
May 19 at 14:43
• @angryavian - Is there any guarantee that max (or sup) and min (or inf) exist? May 19 at 14:47

You can state:$$\forall x\in X\forall y\in Y[xI cannot find a shortcut by means of suprema, infima, maxima or minima. So if that's what you are looking for then this does not answer your question.
• Maybe also $y > \max X$ for all $y \in Y$? May 19 at 14:56
• But there are situations where $X$ does not have a maximum. E.g. if $X=(0,1)$. A maximum of $X$ is by definition an element of $X$.
$$\forall x \in X, \forall y \in Y, x < y$$