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[edit]
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[edit]
- ^ Brown, James Robert. Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures. United Kingdom, Taylor & Francis, 2005. 27.
Well, that’s interesting to know that Psilotum nudum are known as whisk ferns. Psilotum nudum is the commoner species of the two. While the P. flaccidum is a rare species and is found in the tropical islands. Both the species are usually epiphytic in habit and grow upon tree ferns. These species may also be terrestrial and grow in humus or in the crevices of the rocks.
View the detailed Guide of Psilotum nudum: Detailed Study Of Psilotum Nudum (Whisk Fern), Classification, Anatomy, Reproduction