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Surjective, not injective 
Injective, not surjective 
Bijective 
Not surjective, not injective 
When X and Y are both the real line R, then a bijective function f: R → R can be visualized as one whose graph is intersected exactly once by any horizontal line.
If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Generalising this to infinite sets leads to the concept of cardinal number, a way to distinguish the various infinite sizes of infinite sets.
Consider the function f: R → R defined by f(x) = 2x + 1. This function is bijective, since given an arbitrary real number y, we can solve y = 2x + 1 to get exactly one real solution x = (y − 1)/2.
On the other hand, the function g: R → R defined by g(x) = x^{2} is not bijective, for two essentially different reasons. First, we have (for example) g(1) = 1 = g(−1), so that g is not injective; also, there is (for example) no real number x such that x^{2} = −1, so that g is not surjective either. Either one of these facts is enough to show that g is not bijective.
However, if we define the function h: R^{+} → R^{+} by the same formula as g, but with the domain and codomain both restricted to only the nonnegative real numbers, then the function h is bijective. This is because, given an arbitrary nonnegative real number y, we can solve y = x^{2} to get exactly one nonnegative real solution x = √y.
See also: Injective function, Surjection
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