What does $\epsilon = \min\{a-c, d-a\}$ mean?

Question

Can somebody translate the expression $\epsilon = \min\{a-c, d-a\}$ to plain English. Context: Mathematical Analysis, proof that an open interval is an open set.

Answer 1

$\epsilon$ is the smaller of the numbers $a-c$ and $d-a$.

Answer 2

It means that if $a-c < d-a$, then $\epsilon = a-c$, otherwise $\epsilon = d-a$. I.e., $\epsilon$ is the minimum of the set $\{a-c, d-a\}$, or in other words, the smaller of the two numbers.

Answer 3

Imagine some fixed open interval, say $(c, d) = (3, 7)$. Given any $a \in (3, 7)$, we want to come up with some radius $\epsilon$ such that the neighbourhood $(a - \epsilon, a + \epsilon)$ of $a$ is completely contained inside $(3, 7)$. This is obtained by determining which endpoint ($c$ or $d$) that $a$ is closest to. For example:

• If $a = 3.1$, then we should take $\epsilon = a - c = 0.1$, since $(3, 3.2) \subseteq (3, 7)$.
• If $a = 4.9$, then we should take $\epsilon = a - c = 1.9$, since $(3, 6.8) \subseteq (3, 7)$.
• If $a = 5.1$, then we should take $\epsilon = d - a = 1.9$, since $(3.2, 7) \subseteq (3, 7)$.
• If $a = 6.9$, then we should take $\epsilon = d - a = 0.1$, since $(6.8, 7) \subseteq (3, 7)$.

More generally, we can summarize the above statements by saying:

Given any $a \in (c, d)$, we should take $\epsilon = \min\{a - c, d - a\}$.

Answer 4

If $a-c \leq d-a$, then $\varepsilon = a-c$; $~$ If $d-a \leq a-c$, then $\varepsilon = d-a$.