Function

def. Function $f:A\to B$ is a relation between $A,B$ where $\forall a\in A.f(a)$ is defined.

• function must be total, i.e. map everything from $A$ (but doesn’t need to cover all of $B$)
• def. image (=range) of function $f$ is $\text{Im}(f):=\{f(x)\mid x\in A\}\subseteq B$
• def. domain is $A$
• def. codomain is $B$

def. Surjective. If the image of the function is equal to the codomain, it is surjective. Consider a non-surjective function. Visualization.

def. Injective. If $\nexists x_1,x_2\in A$ where $f(x_1)=f(x_2)$ then $f$ is injective. Consider a non-surjective function. Visualization.

def. Invertiable (=bijective). if $f:A\to B$ is both surjective and injective, then $f$ is invertible.

def. Isomorphism of sets. if for set $A,B$ exists $f:A\to B$ which is invertible, then $A,B$ are isomorphic.