$(a,a]=$? Where $a \in \Bbb R$.

Question

We were taught that $[a,a]=\{a\}$ and $(a,a)=\emptyset $, For $a \in \Bbb R$.

So I wonder what will be the result in the case $(a,a]$?

Let $A=(a,a]$.

Then $a \in A$ and $a \notin A$. This is absurd. So I guess $A$ is undefined.

Answer 1

According to the (very reasonable) definition given on Wikipedia:

$$(a,a]:=\{x\in \Bbb{R}:a<x\le a\}=\emptyset$$

I think this is a fair definition.

Answer 2

$A =(a,a]$ is the empty set

Fred

Answer 3

$A=(a,a]=?$

Let $x\in A\implies a<x\le a$ .Hence $A=\emptyset$