Is $(a,a]=\{\emptyset\}$?

Let $a \in \mathbb{R}$, and consider the half open interval $(a,a]$.

Is it correct to write this half open interval as $(a,a]=\{\emptyset \}$? Or $(a,a]=\{a \}$?

• $(x,y]:=\{r \in \mathbf{R}: x<r\le y\}$. Hence it is empty. – Paolo Leonetti Aug 4 '15 at 14:25
• It is not the set containing the empty set, it is just the empty set – BadAtMaths Aug 4 '15 at 14:27
• $\varnothing\neq\{\varnothing\}$. – Asaf Karagila Aug 4 '15 at 14:32
• Possible duplicate of Is $[a, a)$ equal to $\{a\}$ or $\varnothing$? – BCLC Nov 22 '15 at 22:42

No, $(a,a]$ has no elements. It is empty. There is no real number $x$ such that $a < x \le a$. But you do not write that as $\{\varnothing\}$. You write it as $\varnothing$. See the difference?

• I did not know this difference was important, so thanks. So I can clarify, $\{ \emptyset \}$ is the set containing the empty set; is this any set containing the empty set? – möbius Aug 4 '15 at 14:37
• @möbius THE set containing the empty set. There "is" only one, since if two sets have the same elements, they are the same set. – 5xum Aug 4 '15 at 14:39
• @möbius No. it is the set with one element, and this element is the empty set. – Crostul Aug 4 '15 at 14:39

No. Because $\emptyset$, the empty set, is not a real number, i.e. it is not an element of $\mathbb R$, the set $\{\emptyset\}$ is not a subset of $\mathbb R$. ON the other hand, $(a,a]$ is a subset of $\mathbb R$, so there can be no equality.

You are close, though, since $(a,a]$ is the empty set, so $(a,a]=\emptyset=\{\}$.

• I agree that $\varnothing$ is not a real number. But (in some formulations) it may be. Those weird model theorists start with zero as $\varnothing$ and then build up the number system from that... – GEdgar Aug 4 '15 at 14:44
• Only the natural number zero is the empty set; the real number zero is either a Dedekind cut or an equivalence class of Cauchy sequences, and in either case the real number zero is not the empty set. @GEdgar – Carl Mummert Aug 9 '15 at 16:11
• @CarlMummert In the standard model of real numbers. It is possible to construct a model of reals that has empty set as zero. – 5xum Aug 9 '15 at 18:14
• @5xum: well, we can always arbitrarily make any particular number be any particular set. But the usual two constructions of the reals - via Dedekind cuts or via equivalence classes of Cauchy sequences - do not make any real number be represented by the empty set. The real number 0 is not represented in the same way as the natural number 0. – Carl Mummert Aug 9 '15 at 18:30