For instance, in group theory, the term well-defined is often used when dealing with cosets, where a function on a coset space is often defined by choosing a representative: it is then as important that we check that we get the same result regardless of which representative of the coset we choose as it is that we always get the same result when we perform arithmetical operations (e.g., whenever we add 2 and 3, we always get the answer 5). f(x1)=f(x2) whenever x1~x2, then the definition makes sense, and f is well-defined on X/~. Although the distinction is often ignored, the function on X/~, having a different domain, should be viewed as a distinct map . In this view, one says that is well-defined if the diagram shown commutes. That is, that f factors through π, where π is the canonical projection map X → X/~, so that .
As an example, consider the equivalence relation between real numbers defined by θ1~θ2 if there is an integer n such that θ1-θ2 = 2πn, where π (not italicized) denotes Pi. The quotient set X/~ may then be identified with a circle, as an equivalence class [θ] represents an angle. (In fact this is the coset space R/2πZ of the additive subgroup 2πZ of R.) Now if f:R→R is the cosine function, then is well-defined, whereas if f(θ) = θ then is not well-defined.
Two other issues of well-definition arise when defining a function f from a set X to a set Y. First, f should actually be defined on all elements of X. For example, the function f(x) = 1/x is not well-defined as a function from the real numbers to itself, as f(0) is not defined. Secondly, f(x) should be an element of Y for all x∈X. For example, the function f(x) = x2 is not well-defined as a function from the real numbers to the positive real numbers, as f(0) is not positive.
A set is well-defined if any given object either is an element of the set, or is not an element of the set.
US Patent Issued to the Regents of the University of California on July 3 for "Well-Defined Donor-Acceptor Rod-Coil Diblock Copolymer Based on P3HT Containing C 60" (California Inventors)
Jul 09, 2012; ALEXANDRIA, Va., July 9 -- United States Patent no. 8,211,996, issued on July 3, was assigned to The Regents of the University of...