Function
It's a mathematical object responsible for transforming objects from a set A to a set B, preserving their identities. We say
$$f : A \to B$$
to indicate a function that takes elements from a set A and transforms them into elements of a set B. Preserving the identity of the objects in A means that, if we take two elements of A and they are the same, a function from A to B must send them to an unique object in B. In mathematical chattering:
$$\forall x_1,x_2 \in A, x_1 = x_2 \Rightarrow f(x_1) = f(x_2)$$
One-to-one functions
When a function is one-to-one, it preserves the difference of the objects in A when transforming into objects of B. It means that this kind of function guarantees the following property:
$$\forall x_1,x_2 \in A, x_1 \neq x_2 \Rightarrow f(x_1) \neq (x_2)$$
Onto functions
An onto function is capable of producing the set B, because all of the $y \in B$ has a correspondence in A; it means that
$$\forall y \in B, \exists x \in A, f(x) = y$$
Bijective functions
When a function is both one-to-one and onto, we say that it's bijective. In essence, when there exist such a function from a set A to B, we say that A and B are equivalent or that the set A can be rewritten as the set B, and vice versa. It's a perfect translation process. It's like writing a number from a base X to a base Y.
Important functions
Constant function
A function $f : A \to B$ is said to be constant if this property holds:
$$\forall x \in A, f(x) = b, b \in B$$
Identity function
A function $Id_A : A \to A$ of a set A sends every element of A to itself. It means that
$$\forall x \in A, Id_A(x) = x$$
Singleton
A singleton $f_b : \{a\} \to B$, with $\{a\}$ being any unit set, sends the element $a$ to an element $b \in B$. In other words, $f_b(a) = b \in B$. For a better understanding, think of a singleton as a function related to an element of B, with the purpose of pointing to $b$ no matter what it receives. Note that every element of $B$ has a an associated singleton.
Projection
Given a cartesian product $A \times B$, a projection extracts one element of the ordered pairs. So, in this case, there are two projections:
$$\pi_1 : A \times B \to A, \pi_1( \langle a,b \rangle ) = a$$
$$\pi_2 : A \times B \to B, \pi_2( \langle a,b \rangle ) = b$$
The set of all total functions from A to B
It's denoted by $B^A$ or $[A \to B]$.
