Special case

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 a^x + b^y = c^z 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 ${\displaystyle a^{p}\equiv a{\pmod {p}}}$ " is a special case of Euler's theorem, which states "if n and a are coprime positive integers, and ${\displaystyle \phi (n)}$  is Euler's totient function, then ${\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}}$ ", in the case that n is a prime number.
• Euler's identity ${\displaystyle e^{i\pi }=-1}$  is a special case of Euler's formula which states "for any real number x: ${\displaystyle e^{ix}=\cos x+i\sin x}$ ", in the case that x = ${\displaystyle \pi }$ .

References

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