If we define $\mathrm{null}$ as nothing, is it correct to say $\emptyset = \{\mathrm{null}\}$?

Question

Let $\mathcal{X} = \{x\}$. Can we say $\mathcal{X} = \emptyset$ when $x = \mathrm{null}$? Thanks very much!

PS: $\mathrm{null}$ means nothing.

Answer 1

The clear issue is that there is no way in classical mathematics to have an object that can be "nothing". It is not possible to have some $x$ that can "not exist" in some contexts.

This is unlike some programming languages where you can have a variable $x$ that gets instantiated to NULL, and then a list $[x]$ will be the empty list.

This is not a behaviour we allow in mathematics: something must be something, not nothing, so any set $X=\{x\}$ will contain one element (namely $x$) and therefore cannot be empty.

Answer 2

In case your question originates in a kind of perplexity regarding the word " nothing", it may be helpful to observe that modern logic uses quantification to translate this expression into symbolic language, which shows that " nothing " is not a name , and does not denote anything ( such as " nothingness" or " emptyness") .

For example, " there is nothing in box B " reads as

there is no $x$ such that $x$ is in box B.

Hence, logic helps to avoid the traps of ( ordinary language ) grammar. Though in ordinary language, we could say that "nothing is an element of the empty set", a sentence deceivingly suggesting that there is an entity , namely " nothing" , that is the subject of the predicate " being an element of the empty set" , logic tells us that our proposition simply means that :

the open sentence "$x$ is an element of $\emptyset$ " is false for every value of $x$.

Answer 3

• $\mathcal{X} = \{x\}$

Without checking what $x$ is , we can see that the Number of elements in $\mathcal{X}$ is $1$.

In case of Null Set , $\emptyset$ , the Number of Elements is $0$.

Immediately , we can say $\mathcal{X}$ is not $\emptyset$.

In Current Case , $\mathcal{X}$ is a Set having $1$ Element , where that Particular Element is $NULL$.

Update , in response to Query by OP :

There may be Cases where we have to write "Something" like the given Example even though the Null Set may be a Possibility.
Then this is a way to get that :

• $\mathcal{X} = \{x : P(x)\}$

When Condition $P(x)$ is satisfied by Certain Elements , we get $\mathcal{X}$ with those Elements. In Case no element satisfies Condition $P(x)$ , we get Null Set.

Specific Examples :

• $\mathcal{X} = \{x : x^2=+4\}$ will give $\mathcal{X} = \{+2,-2\}$ in Integers
• $\mathcal{X} = \{x : 4-letter-words\}$ will give $\mathcal{X} = \{NOON,NULL,MOON,....\}$ containing hundreds of English words.
• $\mathcal{X} = \{x : x^2=-4\}$ (In real numbers) will give $\mathcal{X} = \{\}$ or Null Set
• $\mathcal{X} = \{x : x+1=x+2\}$ (In real numbers) will give $\mathcal{X} = \{\}$ or Null Set

In no such Case , we can write $\mathcal{X} = \{x\}$ to indicate Null Set.
That always indicates Some Set with Exactly 1 Element.

Update , in response to New Query by OP :

Even when using $\mathcal{X} = \{\uparrow\}$ , it has 1 Element.
$\mathcal{X} = \{\uparrow,\downarrow,\Uparrow,\Downarrow\}$ has 4 Elements.

I think I get what you want. You want a way to put something between the braces yet still say it is the NULL Set.

Using Current Notation , there is a way to achieve what you want :
$\mathcal{X} = \{,\} = \{,,\} = \{,,,\} = \{,,,,\}$ has No Elements , though it looks Weird !

Likewise , you can make your Own Definition , or rather Own Notation , out of the given list , to say that $\mathcal{X} = \{\uparrow\} = \{ \dot{ }\} = \{\cdot\} = \{\Uparrow\} = \{^\circ\}$ indicates the NULL Set. That is your Choice when making your Article.