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It is an adverb, used to form a negative, the opposite of a positive. "The sandwich is tasty." is a positive statement. "The sandwich is not tasty." is the negative of the first statement.
Yes. The eye colour of a classroom of children will have a mode that is a colour, not a number. Their favourite food won't be a number either.
empty set is a set because its name indicate as it is the set.
Empty set or null set
There is only one empty set, also known as the null set. It is the set having no members at all. It is a subset of every set, since it has no member that is not a member of any other set.