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The vertical line test, shown graphically. The abscissa shows the domain of the (to be tested) function. In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x.
A relation that is functional and total. For example, the red and green relations in the diagram are functions, but the blue and black ones are not. An injection [d] A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective.
In general, implicit curves fail the vertical line test (meaning that some values of x are associated with more than one value of y) and so are not necessarily graphs of functions. However, the implicit function theorem gives conditions under which an implicit curve locally is given by the graph of a function (so in particular it has no self ...
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Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output The red curve is the graph of a function, because any vertical line has exactly one crossing point with the curve. A function f from a set X to a set Y is an assignment of one element of Y to each element of X.
In chemistry, the vertical line is used in cell notation of electrochemical cells. Example, Zn | Zn 2+ || Cu 2+ | Cu Single vertical lines show components of the cell which do not mix, usually being in different phases. The double vertical line ( || ) is used to represent salt bridge; which is used to allow free moving ions to move.
In 1998, the Casio fx-991W model used a two-tier (multi-line) display and the system was termed as S-V.P.A.M. (Super V.P.A.M.). The model featured a 5×6-dot LCD matrix cells on the top line of the screen and a 7-segment LCD on the bottom line of the screen that had been used in Casio fx-4500P programmable calculators. [1]
The test functions used to evaluate the algorithms for MOP were taken from Deb, [4] Binh et al. [5] and Binh. [6] The software developed by Deb can be downloaded, [7] which implements the NSGA-II procedure with GAs, or the program posted on Internet, [8] which implements the NSGA-II procedure with ES.