What does this symbol $\wedge$ mean in this context?

Question

I am reading real analysis book and encountered this symbol $\wedge$ and $\vee.$

The author says following:

1. $f\wedge g=\frac{1}{2}(f+g- |f-g|)$,
2. $f\vee g=\frac{1}{2}(f+g+|f-g|)$.

What are the meanings of $\wedge$ and $\vee$? For example, I want to know what $f\wedge g$ means like $f^{+}$ means the positive part of the function.

Edit: he also asserts that $\chi_{A\cap B}=\chi_A\wedge \chi_B$ and $\chi_{A\cup B}=\chi_A\vee\chi_B.$

Answer 1

Let $f(x)$ and $g(x)$ be functions with domain $X\subseteq \mathbb R$.

Define a new function $h=f\wedge g$ by $h(x)=\frac12(f(x)+g(x)-|f(x)-g(x)|)$ for all $x\in X$.

At each $x$, $h(x)$ is the minimum of $f(x)$ and $g(x)$,

because if $f(x)\ge g(x)$ then $|f(x)-g(x)|=f(x)-g(x),$

so $h(x)=\frac{1}{2}(f(x)+g(x)-(f(x)-g(x)))=g(x)$,

whereas for $x$ such that $f(x)\lt g(x),\;$ $|f(x)-g(x)|=g(x)-f(x),$

so $h(x)=\frac12(f(x)+g(x)-(g(x)-f(x)))=f(x)$.

A similar argument shows that $f\vee g$ is the maximum of $f$ and $g$.

Answer 2

The notation $a\wedge b$ stands for $\min\{a,b\}$ and the notation $a\vee b$ stands for $\max\{a,b\}$.

Answer 3

As has already been noted, $\land$ is used to indicate the minimum of $x,\,y$, and $\lor$ their maximum. More generally, the infimum of a set $S$ is denoted $\land S$ and called the meet of $S$, while the supremum of $S$ is denoted $\lor S$ and called the join of $S$.

The symbols $\land,\,\lor$ are also used in propositional calculus to denote and and (inclusive) or. (The indicator function relations in your edit use these meanings for the operators.) If we impose the ordering False $\lt$ True on propositions, $p\land q$ returns the minimum of $p,\,q$ in this sense, and $p\lor q$ their maximum. The theory of semilattices is relevant here.

In LaTeX, you chose to denote the meet symbol as \wedge, but \land would have worked too. The reason this symbol can be called two ways is because it's also used for wedge products in exterior algebras. I'm not aware of any meaning for $\lor$ in interior algebras, except in their modal logic.