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I wanted to bounce my thoughts off some of you to see if I am on the right page. I want to identify the empty sets.

$\{z: \text {z is a horse and z has 6 legs}\}$

I am tempted to say that this is an empty set because no horse has six legs (hopefully) but almost feel like this is incorrect.

$\{n \in \mathbb{N}: n^2 -n + 41 \text{ is not prime}\}$ I want to say that this is NOT an empty set set because $41^2-41+41$ is prime.

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    Correct your final sentence to "*because $41^2-41+41$* **is NOT prime**" so therefore $41$ is in fact an element of the set (*among many others*) – 2017-02-28
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    Looks good to me. I think the key issue with your empty example is that it isn't *entirely inconceivable* that no six-legged horse exists, but don't let that stop you from using that example. It is fine, in spirit. – 2017-02-28
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    I'm sure that in the context of the problem, the set of 6-legged horses was intended as the empty set, from a common knowledge perspective (but of course, although you're sure it's true, you couldn't prove it). One point though -- there is only one empty set, so the better phrasing is "the empty set" rather than "an empty set". – 2017-02-28
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    A less debatable example would be "the set of pigs that can fly". – 2017-02-28
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    @quasi Very good point. That was an oversite on my part. – 2017-02-28

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Both answers are correct, if we assume that no horses with $6$ legs exist (as I'm sure there will be pathological examples of horses with $6$ legs).

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    I believe the mythical Sleipnir ridden by Odin had 6 legs though...[correction, it had 8] – 2017-03-01