Is S^{c} the key difference in this logical equivalence?

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SUMMARY

The discussion centers on the logical equivalence involving the set complement S^{c}. The original statement, ( ∀x ∈ S ∀y P(x) ) <=> ¬( ∃x ∈ S^{c} ∃y ¬P(x) ), is confirmed to be equivalent to the modified statement ( ∀x ∈ S ∀y P(x) ) <=> ¬( ∃x ∈ S ∃y ¬P(x) ). Participants agree that the key difference lies in the presence of S^{c}, which is crucial for understanding the logical relationship between the two expressions.

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Is it equivalent?
<br /> ( \forall x \in S \forall y P(x) ) &lt;=&gt; \neg ( \exists x \in S^{c} \exists y \neg P(x) )<br />

Attempt at solution
I think it should be
<br /> ( \forall x \in S \forall y P(x) ) &lt;=&gt; \neg ( \exists x \in S \exists y \neg P(x) )<br />
The diiference to the above statement is S^{c}.
 
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Yes, your statement is correct.
 


HallsofIvy said:
Yes, your statement is correct.

Please, can you clarify whether it is the first statement or my attempt at solution?
 

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