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I am trying to solve the following problem in Ted Sundstrom's Mathematical Reasoning, Writing and Proof:

What is the truth set of the sentence $(\exists t \in \mathbb{R})(t \cdot x=20) $?

Earlier in the book the following definition is provided:

The truth set of an open sentence with one variable is the collection of objects in the universal set that can be substituted for the variable to make the predicate a truth statement

The variable $t$ is not really a variable. It is already quantified by the existential quantifier and so gets substituted into the predicate "by definition". This is not the case for the variable $x$ and for the sentence to become a statement it must take on a value.

If I had to define the truth set for $x$, I'd probably write something like the following:

$$x \in \left\{ \frac {20} {t} \mid t \in \mathbb R \right\}$$

What do you think? Is this the correct answer? How about the notation? Am I getting it right?

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    $\begingroup$ In your statement $t$ is a bounded variable and $x$ is free They are both variables. Your statement is not a sentence because $x$ is free. Sentences have no free variables. $\endgroup$
    – John Douma
    Apr 7, 2021 at 15:59
  • $\begingroup$ The author would argue the opposite, namely that this is an open sentence and not a statement. The author defines statements as sentences that can either be true or false. Open sentences have a subset of the universal set called the truth set for which they evaluate to true. $\endgroup$
    – user32882
    Apr 7, 2021 at 16:05
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    $\begingroup$ You may want to consult other books. The terminology I use is pretty standard. $\endgroup$
    – John Douma
    Apr 7, 2021 at 16:08
  • $\begingroup$ Um... I'm going to go with the terminology of the book. Thanks. $\endgroup$
    – user32882
    Apr 7, 2021 at 16:08
  • $\begingroup$ @user32882 Then you will have a lot of trouble communicating with people who use the standard terminology. $\endgroup$ Apr 9, 2021 at 12:32

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Your proposed answer calls for division by $0$.

The truth set is just the set of all nonzero real numbers, since for every nonzero $x$ there is a $t$, namely $20/x$, that makes the assertion true.

In set builder notation the truth set is $$ \left\{ x \ | \ x \ne 0 \right\} $$ assuming the universe is the set of real numbers, $$ \left\{ x \ | \ x \in \mathbb{R} \text{ and } x \ne 0 \right\} $$ if you need to make that explicit.

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  • $\begingroup$ how would you express this in set builder notation? $\endgroup$
    – user32882
    Apr 7, 2021 at 15:56
  • $\begingroup$ @user32882 See my edit. $\endgroup$ Apr 7, 2021 at 16:00

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