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What is the exact difference between ∃x∀y(condition) and ∀y∃x(condition)? Translating these into English,

  • ∃x∀y(condition) = There is an x for all y such that (condition) is satisfied.
  • ∀y∃x(condition) = For all y, there is an x such that (condition) is satisfied.

These two seem to be different on the surface, but I could not grasp a vivid understanding of it. Also, I would want to ask how the difference in orders of quantifiers impact the final result.

Centauri_42
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  • Pointwise vs. uniform convergence, for example. – Sean Roberson Oct 19 '21 at 13:25
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    "For every natural number $n$ there is a greater one" vs "There is a natural number that is greater than every natural number" – Mauro ALLEGRANZA Oct 19 '21 at 13:26
  • The statement $\exists x , \forall y , (x+y=0)$ is false. It states that there is a fixed number $x$ that has the property that for all $y$, $x+y=0$. On the other hand, the statement, the statement $\forall y , \exists x , (x+y=0)$ is true, as in this case the value $x$ is allowed to depend on the value of $y$. So we can let $x=-y$. In general, the variables introduced by quanitifers are only allowed to depend on the variables that are introduced earlier in the statement. – Joe Oct 19 '21 at 13:26
  • The former: “There’s a chicken in every pot.” The latter: “There’s one particular chicken that is in every pot simultaneously.” Clearly one statement is much stronger, and implies the other. – mjqxxxx Oct 19 '21 at 13:32

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You have translated the sentence $\exists x \forall y$ (condition) incorrectly.

It means: there exists an $x$ such that for all $y$ (condition). This means that there is just this on single $x$ that works for all $y$ simultaneously.

The sentence $\forall y \exists x$ (condition), means that I can choose a different $x$ for every $y$.

Peter
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    Ah, I see what you are saying, you are interpreting his grammar as similar to the statement "there is a time for every thing" . I think he probably didn't mean it that way, but it is worth asking about. – DanielV Oct 19 '21 at 13:36
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The first condition is stronger then the second one.

An illustration: every person has a mother (his own mother), but there's no a mother of every person.

Botnakov N.
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