Special case

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