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More generally, the restriction (or domain restriction or left-restriction) of a binary relation between and may be defined as a relation having domain , codomain and graph ( ) = {(,) ():}. Similarly, one can define a right-restriction or range restriction R B . {\displaystyle R\triangleright B.}
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.
The corresponding direct system is (F(U), r U,V) where r is the restriction map. The direct limit of this system is called the stalk of F at x, denoted F x. For each neighborhood U of x, the canonical morphism F(U) → F x associates to a section s of F over U an element s x of the stalk F x called the germ of s at x.
However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother of the woman y"; its transitive closure does not relate a woman with her paternal grandmother. On the other ...
The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red). In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space.
For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function. In the study of several complex variables, the definition of a domain is extended to include any connected open subset of C n.
If R is an integral domain, then R[t] is also an integral domain; its field of fractions is the field of rational functions. If R is a Noetherian ring, then R[t] is a Noetherian ring. If R is a unique factorization domain, then R[t] is a unique factorization domain. Finally, R is a field if and only if R[t] is a principal ideal domain.