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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. 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. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.
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.