# Notation for the smallest number in a set?

Let's assume I have a set like $S = \{2,1,3,4,8,10\}$.

What's the math notation for the smallest number in the set?

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$\min(s)$ is what I would expect to see, given that the set has a smallest number. – qaphla Jul 15 '14 at 21:14

The notation you looking for is: $$\min$$

Suppose you have a ordinary finite set $A=\{a_1,\ldots,a_k\}$, then you can write the minimum notation as follows:

$$\min\{a_1,\ldots,a_k\}$$

$$\min\{2,1,3,4,8,10\}=1$$

In case of functions, you can represent its minimum over a set as follows: $$\min_{x\in S}f(x).$$

An example:

$$S=\mathbb{R},\ f(x)=x^2\Rightarrow \min_{x\in S}f(x)=0.$$

Note that a set being infinite certainly does not mean that $\min$ becomes invalid. Consider for example the natural numbers, which certainly have a least element. In the cases where $\min$ is not well-defined and $\inf$ would be more commonly used, one cannot really speak of the least element of a set, and so $\inf$ seems not applicable to this question. – qaphla Jul 15 '14 at 21:22
I don't think anyone does write “$\displaystyle{\min_{x\in S}}$”. I would more readily believe “$\displaystyle{\min_{x\in S} f(x)}$”. Can you point to a published example of this? – MJD Jul 15 '14 at 21:53
In general for a given set $S$ which is nonempty and a subset of an ordered field we define the smallest element in the set to be the element $x \in S$ such that $x\leq y, \ \forall y \in S$. Since you said in a set, I will not introduce the notion of inf. I hope this helps.