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?

Solution 1:

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:

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

In your case,

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

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

An example:

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

Look the comments above for more informations.

Solution 2:

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.