I'm reading Section. The Rational Numbers in textbook Analysis I by Amann/Escher. Below is Proposition 10.9:
and part of its proof:
From what the authors write, I guess that $\color{blue}{\varepsilon \leq 1 \wedge s}$ means $(\varepsilon \leq 1) \wedge (\varepsilon \leq s)$.
My questions:
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Is my understanding correct?
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Is that kind of notation common in mathematics community?
Thank you for your help!
Best Answer
The $\land$ symbol refers to the "meet" of the two numbers. Given a partially ordered set, $x \land y$ refers to the greatest lower bound of $x$ and $y$. That is, it refers to an element $z$ such that $z \le x$ and $z \le y$, and given any $z' \le x$ and $z' \le y$, we also have $z' \le z$. Note that, in a partial order (in particular, if $x$ and $y$ are not comparable), such a point need not exist, and if it does, it need not be equal to $x$ or $y$ (it is, however, unique if it does exist).
In this case, we have a totally ordered set. Therefore, $x \land y$ refers to $\min \{x, y\}$.