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Special case

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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.[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

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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.
  • The unproven Riemann hypothesis is a special case of the generalized Riemann hypothesis, in the case that χ(n) = 1 for all n.
  • Fermat's little theorem, which states "if p is a prime number, then for any integer a, then " is a special case of Euler's theorem, which states "if n and a are coprime positive integers, and is Euler's totient function, then ", in the case that n is a prime number.
  • Euler's identity is a special case of Euler's formula which states "for any real number x: ", in the case that x = .

References

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  1. ^ Brown, James Robert. Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures. United Kingdom, Taylor & Francis, 2005. 27.