Function

Topics: Discrete Mathematics - Calculus

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(definition)

A relation $F$ of a set $A$ on a set $B$ is a function if the following conditions are satisfied:

1. $\text{dom}(F)=A$
2. $(a,b_1)\in F\wedge (a,b_2)\in F⟹b_1=b_2$

We denote such a relation with $F:A\to B$. Functions have a range, an image and a preimage.

Example

Let:

$$F=\{(x,y)\in \mathbb{R}\times \mathbb{R}\mid y=x^2\}$$

$$G=\{(x,y)\in \mathbb{R}\times \mathbb{R}\mid x=y^2\}$$

Notice that $F$ satisfies the conditions, so $F$ is a function, but $G$ does not, so $G$ is not a function.

Vectorial functions are a specific type of functions.