How should logic notations be used when solving an equation?

Question

This is what I found in one of my books (High school books; so probably not very mathematically accurate):

I believe that the calculation itself is accurate; however, regarding the logic notations used, I'm not so sure.

I don't believe that a disjunction symbol should have been used in the last statement (I'm not even sure if using a conjunction symbol is appropriate here either), and I also don't think that the statement has the proper form. I think that the last statement should be replaced by:

Please tell me if I'm wrong or right (if I'm wrong at any point, please clear things up for me)

Answer 1

I disagree with both the book and you.

The initial statement starts with $\exists x \in \mathbb{R}$ , i.e. it claims that there is some $x$ with some property. However, both the books final statement, as well as yours, have dropped that existential quantifier, and end up merely making a claim about $x$. In fact, in logic, that is best seen as a formula with a free variable $x$, rather than a statement with a truth-value.

So, I would say the last statement should be:

$$\exists x \in \mathbb{R} (x = 4 \lor x = 144)$$

Answer 2

From a logic point of view, it is all correct until the last point.

"The existence of some $x$ in $\Bbb R$ such that [blah blah]" is NOT equivalent to $x=4$ or whatever.

If the author is careful enough to repeat the logical symbols wherever they are necessary to be perfectly rigorous, then they should know that reading the last point won't make any sense. What is $x$ in the last point? It was never defined. The $x$ from the points above are "bound" variables that make sense within each statement only.

A correct way to end the argument would be to realize that the existence of $x$ such that $(x-4)(x-144)=0$ is obviously TRUE (and hence so is every statement from the beginning), and that moreover, on a separate note, the set of all such $x$'s is $\{4, 144\}$.