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I have a problem:

Let P be the statement " x $\in$ A and x $\in$ $\mathbb{Z}$ "

Determine the truth value of statement: ($\forall$x)P $\Longrightarrow$ ($\exists$x)P

Is there a set A for which the truth value of the above statement is false? Explain.

My approach is:

The statement is only false when the antecedent is true and the consequent is false.

This is not possible because the antecedent is ... (I have no clue why there is no set A for which the hypothesis ($\forall$x)P of the statement is true).

Can someone help, please?

hanamontana
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  • After "Let $A=\emptyset$" it does not make sense to ask "Is there a set $A$ for which ..." --- No matter what, the statement $(\forall x) P\to (\exists x)P$ is always true because we always assume the universe of discourse to be non-empty – Hagen von Eitzen Mar 05 '21 at 15:27
  • I'm sorry I added the wrong part. There is no set A = $\emptyset$. – hanamontana Mar 05 '21 at 15:31
  • @HagenvonEitzen Can you explain the "universe of discourse" in a simpler way? I'm sorry I can't get it. – hanamontana Mar 05 '21 at 15:33
  • Firstly: it is not very good to change question after given answer - you can write addition explicitly or delete this and open new question etc. And Secondly:$(\forall x)P \Rightarrow (\exists x)P$ is true theorem for every $P$. – zkutch Mar 05 '21 at 15:35

2 Answers2

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Hint: " x ∈ A and x ∈ Z " is false, as $A$ is the empty set.

Wuestenfux
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If $A$ is empty, then every statement of the form

"For all $x \in A$, $x$ has Property $P$"

is vacuously true, even if Property $P$ is something impossible or logically inconsistent, for the simple reason that $A$ contains no counterexamples (because it's empty).

Meanwhile, every statement of the form

"There exists $x \in A$ such that $x$ has Property $P$"

is false, because there does not exist $x \in A$, period.

The conditional statement in the original post has the hypothesis

"If for all $x \in A$, $x$ has Property $P$..."

and the conclusion

"...then there exists $x \in A$, such that $x$ has Property $P$."

A conditional statement is true except when the hypothesis is true and the conclusion is false (i.e. its truth value is defined to be "true, unless we can give a counterexample"). If $A$ is empty, what truth values does that assign to the hypothesis, the conclusion, and the conditional statement as a whole?

Rivers McForge
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