A "case" can be built for either side of an opinion. The concern is which case might prevail in a Court of Law.
In my humble opinion, your Property Manager is the wiser of the two. Discertion is the better part of valor. I would find a way to dance around the issue linguistically and try to avoid using those words.
Does anyone know the truth?
In a word . . . . No !
It's a mite humorous that it was your Property Manager who suggested that you attend the Fair Housing Seminar where you were taught a point of view adverse to the opinion that prompted your attendance.
Actually, your cards would appear to discriminate against "Single Parent Households", IF Children means the same as "Kids", and the Rate is the same for 2 Parent Households as it is for Single Parents. As a LandLord, I would always prefer having the potential additional supervision that might be present with two parents.
You could consider just saying "No more that 4 Human Beings, one of whom must be above the age necessary to sign a Legal Agreement." But then you are equating One Adult with 3 Children to the equivalent of Two Adults and 2 Children. How about 4 Adults, and No "Kids" but they start spawning shortly after moving in? I don't know if "Adults Only" buildings or compounds are permitted in Nevada? In Vermont, we allow those that require occupants to be "55 or Older" but your children better not plan on any extended visits, even if they're 40 years old.
Or is the lawyer teaching the fair housing class correct?
There is no Right or Wrong until a Court of Competent Jurisdiction decides. And then even that is subject to appeal. Only the Supreme Court will know, and they apparently haven't decided yet . . . . and I don't want to go there.
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