Marmorphism is a concept in mathematics, which describes the characteristics of functions or mappings.

A function is said to be surjective if at least one element in its domain corresponds to each element in its companion domain. In other words, the range of a injective function is equal to its coset.

In mathematics, the surjection property of a function is closely related to the relationship between its definition domain and cotangent domain. Injective, surjective and bijective are three types of functions, which are distinguished according to the different ways of correlation between the definition domain and cotangent domain of the function. Corresponding to surjection is injectivity, which refers to a function that maps different variables to different values. Birefringence is a function of injectivity and surjectivity. Intuitively speaking, bijective functions form a correspondence, and each input value has exactly one output value, and each output value has exactly one input value.

The difference between surjective and injective:

1. Consider a class with 10 students, and each student has a unique student number. Now, if we define a function that maps students' student numbers to their names, then this function is an injective function. Because each student number corresponds to only one student name, each output value (name) corresponds to only one input value (student number).

2. If we define another function to map students' student numbers to their ages, then this function may not be injective. Because there may be multiple students with the same student number but different ages. However, this function is still surjective, because at least one student number can be found corresponding to each age.

3. Suppose we have a function that maps natural numbers to natural numbers. This function is defined as f(x)=2x. This function is not injective or injective. Because there are two output values (f(5)= 10 and f(6)= 12) corresponding to some input values (for example, x=5), it is not injective. Because some output values (such as 14) have no input values to map to, it is not surjective.

4. The main difference between injectivity and surjection is that injectivity emphasizes the one-to-one correspondence between input and output, that is, one output value can only correspond to one input value; On the other hand, surmorphism emphasizes the coverage of the range, that is, the range must contain all the elements of the companion range.