Applications

An application is a rule that given two sets X and Y, each element of X is associated with an element of Y (and only one)

If we call that rule f, this is denoted as: f|X\rightarrow Y

If f is associated x \in X with y \in Y it is denoted y=f(x) (and it is said that y is the image of x), and you can also use the notation x\rightarrow y

X is called domain, and the set f(X)=\{f(x)|x\in X\} is the image or the path of f. (With more rigor we can define it from previous concepts, as a relation f\subset X x Y such that, for everything x\in X there is a single y\in Y that meets (x, y)\in f)

Applications can be of the type:

• f|X\rightarrow Y is called injective \text{Si }f(a)\not=f(b) always that a\not= b (can be denoted in abbreviated form as f(a)=f(b) \Rightarrow a = b)
• f|X\rightarrow Y is called surjective if for each y \in Y there is x\in X such that y=f(x)
• f|X\rightarrow Y is called bijective if is injective and suprayectiva at the same time

Instead of applications, and with the same meaning, there is also talk of functions; commonly, the term function is used when dealing with a mapping between sets of numbers. In some Spanish-speaking countries, the term mapping is also used from the English term map. Sometimes an application is also said to be 1−1 with the meaning of being an injective application