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One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"—if there is a bijection between them.
The game {0|} is called 1, and the game {|0} is called -1. The game {0|0} is called * (star), and is the first game we find that is not a number. All numbers are positive, negative, or zero, and we say that a game is positive if Left has a winning strategy, negative if Right has a winning strategy, or zero if the second player has a winning ...
The function g : R → R defined by g(x) = x 2 is not surjective, since there is no real number x such that x 2 = −1. However, the function g : R → R ≥0 defined by g(x) = x 2 (with the restricted codomain) is surjective, since for every y in the nonnegative real codomain Y, there is at least one x in the real domain X such that x 2 = y.
Conway's Game of Life and fractals, as two examples, may also be considered mathematical puzzles even though the solver interacts with them only at the beginning by providing a set of initial conditions. After these conditions are set, the rules of the puzzle determine all subsequent changes and moves.
In mathematics, an injective function (also known as injection, or one-to-one function [1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f(x 1) ≠ f(x 2) (equivalently by contraposition, f(x 1) = f(x 2) implies x 1 = x 2).
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