What is $\inf\emptyset$ and $\sup\emptyset$?

Question

I was told that $\sup\emptyset=-\infty$ and $\inf\emptyset=\infty$, where $\emptyset$ is the empty set. This seems paradoxical to me as to how supremum can be less than infimum. Is there any proof for this or is it taken for granted ?

Answer 1

In the context of the extended real number line (i.e. real numbers and $\pm \infty$), this is true. It's not hard to prove - the supremum of a set is defined as the least upper bound of the set. Every number* is an upper bound to the empty set (since "upper bound" means "greater than or equal to every element"). So, the least upper bound is the least number - which is $-\infty$. This acts similarly for the infimum.

I agree that it is counterintuitive - it is, indeed, the only case where the supremum is less than the infimum. However, it does follow from definition. One way to think about it is that the supremum of a set $S$ is what we get if we take a point, and drag it down from $\infty$ until it cannot go lower without hitting $S$ and the infimum is what happens if we take a point and drag it up from $-\infty$ until it hits $S$. That is, we sort of imagine $S$ like an impassable block of stuff, and the supremum and infimum are clamped to the sides of it. But if there is no $S$, then there is no block, and as we clamp these points together, they just pass through each other and keep going - they always had motion inwards, but now nothing stops them, so they end up at $-\infty$ and $\infty$ respectively, as far as they can go.

(*"Number" in the sense of "element of the extended real line")

Answer 2

Since every real number $x$ is an upper bound for $\emptyset$, $x \ge \sup \emptyset$ for all $x\in \Bbb R$. Therefore $\sup \emptyset = -\infty$. Similar reasoning gives $\inf \emptyset = +\infty$.

Answer 3

If we look at the definition of supremum:

We say that $x$ is the supremum of a set $S$ if $x$ is the least upper bound of $S$. I.e., $x \geq s$ for all $s \in S$ and $x \leq y$ for any $y$ that is an upper bound on $S$.

So if we consider $\emptyset$, every $x\in \mathbb{R}$ is an upper bound of $\emptyset$. So the supremum of $\emptyset$ must be the $\min(\mathbb{R})$, which is often $-\infty$.

We can reason similarly for the infemum.