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Ghorpade-Limaye, A couse in Calculus and Real Analysis, p.5) says that $\emptyset$ does not have a supremum but there is not any explanation.

My question is: Why?

Martin Sleziak
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4 Answers4

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Well, a supremum of a set $A$ is a number $a$ such that

  • $a$ is $\ge$ every element of $A$ (such an $a$ is called an upper bound of $A$), and

  • $a$ is $\le$ any number $b$ with the previous property.

Now, here's the problem:

What isn't an upper bound of the empty set?

For example, $4$ is an upper bound of $\emptyset$. Why? Well, $4$ is $\ge$ every element of the emptyset. You don't believe me? OK, find me an element of the emptyset which is not $\le 4$.

(Similarly, every element of the emptyset is an elephant.)

So the problem is that every real number is an upper bound of $\emptyset$. But there is no least real number, so the emptyset doesn't have a least upper bound - that is, the emptyset has no supremum.

Noah Schweber
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Every real number is an upper bound for $\emptyset$. So there is no smallest real number. The same reasoning applies for the infimum. What is interesting, however, is that you can use this reasoning to find $\sup\emptyset = -\infty$ and $\inf\emptyset = \infty$ in the extended reals.

G. Snapsmath
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  • I could'nt believe that sup $\emptyset = -\infty$ and vice versa. –  Oct 09 '16 at 17:28
  • Yeah it is definitely counter-intuitive, but let's tackle the supremum. Every real number is an upper bound for $\emptyset$. This means $\sup\emptyset \le x$ for all $x \in\Bbb{R}$. But the only number that satisfies this is $-\infty$. – G. Snapsmath Oct 09 '16 at 17:35
  • Definition of SUP says that Let $S$ be a subset of $\mathbb{R}$. An element $M\in\mathbb{R}$ is called a supremem of the set $S$ if bla bla... So, $-\infty$ or $\infty$ is not a element. –  Oct 09 '16 at 17:49
  • i.e., $M$ could not equal to $-\infty$ or $\infty$. –  Oct 09 '16 at 17:50
  • I could not believe still. –  Oct 09 '16 at 17:51
  • That is the analysis definition of supremum. In general all you need is a partially ordered set. i.e. Let $(P,\le)$ be a partially ordered set, then define supremum in the obvious way. – G. Snapsmath Oct 09 '16 at 17:57
  • What is $-\infty$ mean? Is it a number or a real number or an element? NO! So, How it satisfy $-\infty$? –  Oct 09 '16 at 17:57
  • $-\infty$ is the smallest element in the ordered set $\overline{\mathbb{R}}$. In fact we could extend this result. Let $(P,\le)$ be any partially ordered set with a smallest element $\alpha$. Then $\sup\emptyset = \alpha$. – G. Snapsmath Oct 09 '16 at 18:01
  • What is $\overline {\mathbb{R} }$ mean? –  Oct 09 '16 at 18:11
  • So, $-\infty$ could not be an element since $-\infty$ is not an element but you says that $-\infty$ is the smallest ''element'' in order set.... –  Oct 09 '16 at 18:12
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    $\overline{\mathbb{R}} = \mathbb{R}\cup{-\infty, \infty}$ So basically you just take $\mathbb{R}$ and give it a largest and smallest element. This is known as the extended reals – G. Snapsmath Oct 09 '16 at 18:16
  • Okey, thanks... –  Oct 09 '16 at 18:29
  • In particular, the extended reals form a complete lattice. – MathematicsStudent1122 Oct 10 '16 at 10:22
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Every real number is an upper bound of $\emptyset$. That is to say, the set of all upper bounds of $\emptyset$ is $\Bbb{R}$, which has no least element.

Micah
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The supremum is the least upper bound. What's an upper bound of $\emptyset$? It is $a\in\mathbb R$ such that $x\leq a$ for all $x\in \emptyset$. Since no such $x$ exists, either you interpret that all $a$ are upper bounds, so there is no least upper bound, or that no $a$ is an upper bound, in which case there is no least upper bound either.

Martin Argerami
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