Specific, usually well-known application of a mathematical rule or law
In
logic, especially as applied in
mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a
generalization of A.[1] A
limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. If B is true, one can immediately deduce that A is true as well, and if B is false, A can also be immediately deduced to be false. A
degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.
Examples
Special case examples include the following:
All squares are
rectangles (but not all rectangles are squares); therefore the square is a special case of the rectangle.
Fermat's Last Theorem, that an + bn = cn has no solutions in positive integers with n > 2, is a special case of
Beal's conjecture, that ax + by = cz has no primitive solutions in positive integers with x, y, and z all greater than 2, specifically, the case of x = y = z.