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Onto or Surjective functions
If we have a function f : A→B, such that every element of B is mapped to by at least one element of A the function is called an Onto function or Surjective function. The whole of A is mapped onto the whole of B. This means that B equals the range of f . The codomain is equal to the range.
Properties of Functions. Onto functions or Surjective functions, One-to-One functions or Injective
One-to-One functions or Injective functions
If we have a function f : A→B, such that elements of B is mapped to by at most one element of A the function is called a One-to-one function or Injective function. There is one element in B for each element of A, the elements are in a one-to-one relation. The whole of A is, for each element in an unique way, mapped into B.
Invertible functions or Bijective functions
A function that is both surjective and injective is an Invertible function or Bijective function.
And all three of them
The figures above are illustrations, from left to right, of an Onto, Surjective function, an One-to-One. Injective function and an Invertible, Bijective function.
Properties of Functions. Onto functions or Surjective functions, One-to-One functions or Injective
Let us start with the onto case. As you could see we have no elements left over in the green set ,the range B, but one of the elements is mapped to by two elements in the red set, the domain A
In the one-to-one case no elements are mapped to by more than one element. One element is mapped to by one element.
In the Bijective case all elements in B are mapped by one element and only one element in A, and all elements in A are mapped to one element and only one element of A.
We could use a few continuous functions as examples too. The ranges and codomains are within the white rectangles.
Yet again we have an Onto, Surjective function, an One-to-One. Injective function and an Invertible, Bijective function.
Given just an expression like y=x2, then the question whatever a function is surjective, injective or bijective is as much dependent on the function itself as of the chosen domain and range. Let us for example look at the function y=|x−2|. If the domain and range are the reals then the function is neither of the above. But let us limit the range to the reals equal to or above zero only. We now have a surjective function. If we also choose to limit the range to the reals above 2 our function will not only be injective but bijective too. These cases correspond to the three cases below.
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