Minimum of empty set in $\infty$ why is that?

Question

According to a definition a minimum in a set is the smallest number in it.

My professor claimed: the minimum in empty set in $\infty$ why is that? $\infty$ isn't even a member of the empty set which contradicts the definition, plus it's will make more sense to me if it's $- \infty$

Answer 1

As others have noted, the minimum and maximum of the $\emptyset$ are undefined, since a minimum or maximum must belong to the set, but the infimum and supremum are well-defined.

The infimum of a set is the greatest lower bound. That is, it is the largest number such that no member of the set is smaller than it. Now if we take any real number $x$, it is surely true that no member of $\emptyset$ is smaller than $x$. What is the largest $x$? Well, there's no largest real number, so $\infty$ is a sensible agreement.

Similarly, the supremum of $\emptyset$ is $-\infty$.

Answer 2

As you've written it, your professor is wrong.

However, there is a similar (but definitely different) notion of a greatest lower bound, also called an infimum. This value is defined as the largest value that has no set element larger than it. For an empty set, the greatest lower bound must be $\infty$.

Answer 3

Conventionally, the sum ($\displaystyle\sum$) of no term is $0$ and the product ($\displaystyle\prod$) of no factor is $1$. This way, the result remains coherent if you append an element. $0$ is neutral for addition, $1$ neutral for multiplication.

The same holds with the minimum: the minimum of no term must be plus infinity, so that if you append an element, the new minimum is that element. $\infty$ is neutral for the minimum.

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Similarly, the union of no set should be defined as the empty set, and the intersection of no set, as the universe.

Answer 4

You are right, the definition doesn't apply to the empty set. What your professor should have said is that for the empty set it's defined separately as $\infty$.

Why is this a sensitive definition?

You have some properties that you would like to remain true in the case of the empty set.

For example, if $A \subseteq B$ then $\min A \ge \min B$. That's easy to prove for non-empty sets, so you would like to define it for the empty set so this remains true.

But $\emptyset \subseteq B$ for all subsets $B$, so you'll need $\min \emptyset \ge \min B$ for all subsets $B$. In particular if $B=\{x\}$, $\min \emptyset \ge x$ for all $x \in \mathbb R$. Obviously there's no such real number and this clearly motivates the defintion $\min \emptyset = \infty$.

Other properties involving union, intersection or complement of sets also can be used to motivate this definition and in general all works very nice with this definition.

Note: For subsets of $\mathbb R$ in general one should talk about the infimum, not the minimum, since not all sets have a least element.

Answer 5

Let's consider the empty set $\emptyset\subset \mathbb R$ as a subset of $\mathbb R$.

Every real number $r\in \mathbb R$ is a lower bound of $\emptyset$ and this is because there is no $x\in \emptyset$ such that $x\lt r$. So the set of all lower bounds of $\emptyset$ is $\mathbb R$.

So $\inf \emptyset=\sup \mathbb R=\infty$, where $\infty$ is not a real number rather it's just a symbol to show that $\mathbb R$ is unbounded above.

Using the same arguments, mutatis mutandis, for upper bounds, you will get $\sup \emptyset =\inf \mathbb R=-\infty$.