Find a universe for variables x, y, and z for which the statement ∃x∃y∀z((x = z) ∨ (y = z)) is true and another universe in which it is false.

I'm not sure if I am approaching this question the right way, but the way I think of it is like this: say one universe is all real numbers, then the statement would be true. Now I have to find another universe where this statement is false, but I can't find any universe where this statement would be false. I might be thinking of this question the wrong way though.

  • $\begingroup$ If it were true in the real numbers, what would $x$ and $y$ be? $\endgroup$ – MJD Oct 7 '17 at 4:26
  • $\begingroup$ Why would you think the statement is true for the reals? $\endgroup$ – Graham Kemp Oct 7 '17 at 4:27

The statement $$\exists x\exists y\forall z\;\bigl((x = z) \lor (y = z)\bigr)$$ asserts that there exist elements $x,y$ in the universe such that, for each $z$ in the universe, either $z=x,\;\,$or $z=y,\;\,$or both (i.e., $z$ is equal to one of $x,y$).

Based on that understanding, the statement is true for a given universe if and only if the universe is nonempty and has at most two elements.

If the universe has exactly one element, $a$ say, let $x=y=a$.

If the universe has exactly two elements, $a,b$ say, let $x=a, y = b$.

In both of the above cases, the statement is true, since any $z$ would have to be equal to one of $x,y$ (there's no way $z$ can avoid that).

On the other hand, if the universe has at least $3$ elements, $a,b,c,\;$say, there's no way to choose $x,y$ so that each $z$ is equal to one of $x,y$. If such elements $x,y$ were to exist, the set $\{x,y\}$ would have at most two elements, hence the statement would be false for at least one of the test cases $z=a, z=b, z=c$.

Key point: The order of the quantifiers matters.

For example, if you change the statement to $$\forall z\exists x\exists y\;\bigl((x = z) \lor (y = z)\bigr)$$ the new statement asserts that for each $z\;$in the universe, there exist elements $x,y\;$in the universe such that $x=z,\;$or$\;y=z,\;\,$or both.

But the new statement is true in any universe since, for any choice of $z$, we can simply choose $x=z,\;$and then for $y$, we can choose any element of the universe (e.g., $y=z$).

  • $\begingroup$ By th OPs comment I suspect the confusion comes from ignoring the order of the quantifiers, and actually they reasoned starting from the universal quantifier, i.e. "for all z there exists at least ...". Maybe you could highlight that the order of the quantifiers makes all the difference. $\endgroup$ – Rolazaro Azeveires Oct 7 '17 at 8:57
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    $\begingroup$ Good suggestion. I edited my answer to address that point. $\endgroup$ – quasi Oct 7 '17 at 9:12
  • $\begingroup$ Yeah I was having a lot of trouble with the order of the quantifiers and figuring out if they mattered or not. I really appreciate the help but I'm still a bit confused on why the statement would be true if ∀z was in the front as opposed to the back. $\endgroup$ – Spectre Oct 7 '17 at 20:35
  • $\begingroup$ Because if $\forall z$ was first, then for each $z$, the choice of $x,y$ is allowed to depend on the selected $z$, since once chosen, $z$ is constant. $\endgroup$ – quasi Oct 7 '17 at 20:37
  • $\begingroup$ If $\exists x\exists y$ is first, it's the $x,y$ that must be chosen first, and they can't force the choice of $z$ $\endgroup$ – quasi Oct 7 '17 at 20:39

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