What exactly do you need to do: translate these statements to set notation or explain why the first line and the second line say the same thing?

Let B be the set of male cats and H be the set of white cats. Then the complement of H, i.e., H', is the set of non-white cats, so is the set of male non-white cats. Saying that this set is nonempty is the same as saying that there exists a male cat that is non-white.

In general, there is a direct correspondence between set operations on (the extensions of) properties and logical connectives. Let A be the set of objects having a property P and B be the set of objects having a property Q. In set notation, and . The set A is called the extension of property P and similarly for B and Q. Then , and where , and mean "and," "or," and "not," respectively. So, the laws like De Morgan's carry over from logic to set theory and back: .

It should be clear that H is the extension of the property , i.e., . Using the fact that and the correspondence described above, .

Note that (\cap in LaTeX) denotes the intersection of A and B, i.e., the set of common elements. In contrast, denotes the intersection of the elements of A when A is afamilyof sets. I.e., if , then . A similar thing holds for (\cup in LaTeX) and .