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. It is also a special case of the rhombus. If an isosceles triangle is defined as a triangle with at least 2 identical angles, an equilateral triangle is therefore a special case. (However, this is not true if an authority follows a different linguistic prescription of an isosceles triangle having exactly 2 sides.) 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 a p ≡ a ( mod p ) {\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 ϕ ( n ) {\displaystyle \phi (n)} is Euler's totient function, then a φ ( n ) ≡ 1 ( mod n ) {\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}} ", in the case that n is a prime number. Euler's identity e i π = − 1 {\displaystyle e^{i\pi }=-1} is a special case of Euler's formula which states "for any real number x: e i x = cos x + i sin x {\displaystyle e^{ix}=\cos x+i\sin x} ", in the case that x = π {\displaystyle \pi } .
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