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I just want to make sure I'm negating the following logical statement correctly (for a contradiction proof):

For every set $A$, there exists a well ordered set $V$ such that there exists no surjection $\pi: A \rightarrow V$.

I'm negating this as:

For every set $A$, there exists a well ordered set $V$ such that there exists a surjection $\pi: A \rightarrow V$.

Is this a correct negation?

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2 Answers 2

up vote 4 down vote accepted

No. You've only negated one part of the statement. Negating each part:

There exists a set $A$ so that for all well-ordered sets $V$ there exists a surjection $\pi:A\to V$.

I think I've negated what you're actually going for here.

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Thank you. This is actually exactly what I was hoping it would be. –  Zermie Feb 16 at 7:49

That is not a correct negation. Quantifiers switch when you negate. As such, the "for every" becomes a "there exists...such that", the "there exists" becomes a "for every", and "such that" is deleted for grammar reasons.

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