# How to read this Arg Max definition in plain english

I was reading on Wikipedia about Arg max (https://en.wikipedia.org/wiki/Arg_max) and they gave the following equation. While I get most of this line, how would you read the following in plain English?

$$\underset{x\in S\subseteq X}{\operatorname{arg\,max}}\, f(x) := \{x \mid x\in S \wedge \forall y \in S : f(y) \le f(x)\}$$

I'll start.

We define $\operatorname{arg max}$ for $f(x)$ (for $x$ a member of $S$ which is a subset of real numbers) as the set of values $x$ given $x$ a member of $S$…

It's the big caret that throws me off. How would you finish it? I'm also not sure what $S$ is, just the input domain?

## 3 Answers

The caret (the symbol is $\wedge$, written as \wedge in LaTeX) means "and". The definition says that the argmax is the set of all $x$ which are in $S$ and satisfy $f(y) \leq f(x)$ whenever $y$ is also in $S$. We need this funny definition mainly because the argmax might have more than one element, for example this occurs with $f(x)=-x^4/4+x^2/2$. It also might have no elements, such as $f(x)=x^2$ on $\mathbb{R}$, but this is a more minor issue.

Remark: it would really be better to write it as $\{ x \in S \mid \forall y \in S \: f(y) \leq f(x) \}$. That's because we can't use unrestricted set comprehension, we can only extract the objects in a set which satisfy some property. So you may as well "frontload" that.

• Semantically, it would be cleaner to use \land (short for "logical and") rather than \wedge in this place. – Daniel Fischer Jun 30 '16 at 9:41

Given a set $S$ and a real valued function $f$ then $\operatorname{argmax}_S f = \{ s \in S | f(s) = \sup_{x \in S} f(x) \}$.

That "big caret" is "and". This says "Argmax(f) is the set of all values of x such that f(x) is larger than equal to f(y) for all y in the set S." Equivalently, "Argmax(f) is the set of all x that give the maximum value of f."