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Preliminary Content
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
Loosely speaking, a function is surjective if every point in the codomain actually gets mapped to by some point in the domain. A function will not be surjective if there are points in the codomain that do not get mapped to. Using our constant function from above, we see that f:R→R given by f(x) = 1 is not surjective, since we can find at least one point in the codomain (say, π), that does not get mapped to.
Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Let r,s∈ℙ, and let R,S be r- and s-element sets. Recall that a function is injective (or one-to-one) if each element of the domain maps to a unique element of the range, surjective (or onto) if every element of the range has a preimage, and bijective if it is both injective and surjective.
Linear Operators and Matrices
Published in Wai-Kai Chen, Mathematics for Circuits and Filters, 2000
Cheryl B. Schrader, Michael K. Sain
If R(f) = T, then f is said to be surjective (onto). Loosely speaking, all elements in Tare used up. Iff: S-+ T has the property that f(sJ) = f(s2) implies s, = s2, then f is said to be injective (one-to-one). This means that any element in R(f) comes from a unique element in D(f) under the action of f. If a function is both injective and surjective, then it is said to be bijective (one-to-one and onto).
Computational networks and systems – homogenization of variational problems on micro-architectured networks and devices
Published in Optimization Methods and Software, 2019
Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber
Let , and . Since ϵ is a feasible length of periodicity the domain Ω is covered by pairwise disjoint ϵ-cells by Equation (1). That means, for each there exists a unique cell index with . With we define a point on . Assume that is fixed. Then, takes the same value for all . In other words, depends only on the ϵ-cell with , but not on the actual position of z within the ϵ-cell . It follows, that Instead of , each other element of the ϵ-cell can be used as an argument of and the value of x does not change. In particular, the corner point of the ϵ-cell can be used. Because of for all and each we can consider this point as a representative of the ϵ-cell . The function with is surjective, but not injective. For each we introduce the vector