Surjective mapping, also known as onto mapping, is a function where every element in the output set has at least one corresponding element in the input set. This means that every element in the output set is being mapped to, or covered by, at least one element in the input set. Bijective mapping, also known as one-to-one correspondence, is a function where every element in the output set has a unique corresponding element in the input set. This means that each element in the output set is being mapped to one and only one element in the input set.
To determine if a mapping is surjective, you can check if every element in the output set is being mapped to by at least one element in the input set. If this is true, then the mapping is surjective. To determine if a mapping is bijective, you can check if every element in the output set is being mapped to by exactly one element in the input set. If this is true, then the mapping is bijective.
Surjective and bijective mapping are important concepts in mathematics and computer science. These types of mappings allow for a one-to-one correspondence between elements in different sets, making it possible to perform operations such as inverses and compositions. Bijective mapping is also used in cryptography to ensure that no information is lost or duplicated during encryption and decryption.
An example of a surjective mapping is the function f(x) = x^2, where the input set is all real numbers and the output set is all non-negative real numbers. Every non-negative real number has at least one corresponding element in the input set (for example, 4 is mapped to by both 2 and -2). An example of a bijective mapping is the function g(x) = 2x+1, where the input set is all real numbers and the output set is all real numbers. Every real number has a unique corresponding element in the input set (for example, 7 is only mapped to by 3).
Surjective and bijective mapping are both types of injective mapping, which is a function where no two elements in the input set are mapped to the same element in the output set. While surjective mapping ensures that every element in the output set is being mapped to, bijective mapping goes a step further and ensures that every element in the output set is being mapped to only once. Injective mapping is important in mathematics and computer science for its ability to preserve information and allow for efficient operations.