How to come up with the negation of "all cars are black" or "all cars are black and have four seats"? I would like to come up with the negation of the statements: "all cars are black" and "all cars are black and have four seats". My understanding is that:
Negation of "all cars are black": Some cars are red.
Negation of "all cars are black and have four seats": Some cars are red OR have three seats.
I am a bit confused if the above is correct or not. My understanding of "NOT all cars are black" is that some must be a different color, but I don't know if I can specify exactly what color that might be (ie, red).
Furthermore, my understanding is that the second statement: "all cars are black and have four seats" is that it is of the form $x \ \land \ y$. Hence, is it natural for the negation to be exactly $\neg x \lor \neg y$? 
Thanks!!
 A: you just need to find a car that is not black. There's one parked on my block if you need a photo.
A: Avoiding not is a bad idea and your examples show why.  The negation of "all cars are black" is "not all cars are black", which is equivalent to "some cars are not black".  If you pick a color, like red in your example, nothing tells you that some cars are red.  They might be lots of colors but not red.  Your statement with Furthermore is correct. The negation of  "all cars are black and have four seats" is "not(all cars are black and have four seats)" which is equivalent to "some cars are not black or some cars do not have four seats".  Very few cars have three seats, though a number of them have two.  If we painted every car black and there were no three seaters, the statement "all cars are black and have four seats" would be false, but your suggested negation would be false as well.
A: For the first statement, the negation would be "There exists some car which is not black."
In general, when negating a "for all" statement you should use a "there exists" statement. Conversely, when negating a "there exists" statement you should use a "for all" statement. 
And yes, you've got the negation of the second statement right.
