Is this symbolic statement impossible?

Question

Is this statement logically impossible if x is a single real number (i.e. not a set)?

$$(x<5) \land(x>7)$$

it seems to me that x cannot both be greater than 7 and less than 5 if it is a single number on the numberline

Answer 1

For any $x \in \mathbb{R}$, the statement

$x < 5 \wedge x > 7$

is logically false. Equivalently, the statement

There exists an $x \in \mathbb{R}$ such that $x < 5$ and $x > 7$

is logically false.

Answer 2

Provided that the domain that $x$ varies over is $\mathbb{R}$, and that $<,>,5,7$ all have their normal meanings in this domain, the statement $$(x< 5) \land (x > 7)$$ is not true for any choice of $x$.

I'm not sure you'd want to call this a logical impossibility though, as there is nothing about logic that restricts $x\in \mathbb{R}$, or fixes non-logical symbols to have a particular interpretation.

Answer 3

Yes, you are right, this is impossible.

Though, if you talk about sets :

$$(x<5)\cap (x>7)=\varnothing.$$