# The Empty Set and Identifying It

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.

• 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) – JMoravitz Feb 28 '17 at 21:21
• 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. – The Count Feb 28 '17 at 21:21
• 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". – quasi Feb 28 '17 at 21:29
• A less debatable example would be "the set of pigs that can fly". – quasi Feb 28 '17 at 21:31
• @quasi Very good point. That was an oversite on my part. – frillybob Feb 28 '17 at 21:34

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).