Special case

Last updated November 19, 2024

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

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References

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

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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.

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