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Vector analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Recall that a function f : A → B is injective if f(a) = f(t) implies a = t. An injective function has an inverse: for y in the image of f, we define f−1(y) to be the unique a with f(a) = y. A function f : A → B is surjective (or onto) if the image of A under f is all of B. Suppose f is both injective and surjective. By surjectivity, for each b ∈ B there is an a ∈ A with f(a) = b. By injectivity, this a is unique. This surjectivity guarantees existence of a solution and injectivity guarantees uniqueness of the solution.
Continuity
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Remember that, if f is an injective function, there exists its inverse function f−1. Also, the graphs of f and f−1 are symmetric with respect to the graph of y = x, so if f is continuous, we expect the same for f−1.
Analysis
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
For an injective function f mapping X into Y, there is an inverse function f-1 mapping the range of f into X which is defined by: f-1(y) = x if and only if f(x) = y.
Optimization of reliable cyclic cable layouts in offshore wind farms
Published in Engineering Optimization, 2021
The injective function defines the embedding of the nodes in the plane. Correspondingly, the embedding of an arc is the closed line segment from to , written as . The corresponding open line segment is denoted . The embedding of a path is the concatenation of the embedding of its arcs , and thus defined as . Further, the interior of the polygon bounded by is denoted . A list of symbols introduced is given in Table 1.
On the representations of L-equivalence relations on L-fuzzy sets with applications to locally vague environments
Published in International Journal of General Systems, 2020
-equivalence(resp. equality) relations on a set X are nothing else but -equivalence(resp. equality) relations on the -fuzzy set . Given any , the mappings , , defined by are the smallest and the greatest -equivalence relations on μ. If for all , then will be an -equality relation. will also be an -equality relation when is an injective function.
A New Metatheorem and Subdirect Product Theorem for L-Subgroups
Published in Fuzzy Information and Engineering, 2018
It can be established that is an injective function. Also, is an order reversing function from J into . The ordering of induces a pointwise partial order on . That is for P and Q in Further as is a lattice, is also a lattice. Tom Head enlarged the domain of a binary operation on G to by convolutional extension method. Also, the binary operation on G induces a binary operation on and .