Functions in Math

Definition

The definition of a function is:
- $Γ (f) = {(a, f (a)) : a \in A} \subseteq A \times B$ .
- $f : A \to B, x \mapsto x^{2}$
  - This is the same as: $f (x) = x^{2}$
  - Where the only $x$ input I can put in is whatever is in $A$ . This must produce whatever is in $B$ .
  - $A$ is the Domain and $B$ is the Codomain.
Mathematicians think of a function that consumes elements in $A$ to produce and element in $B$ . This is where Domain and Codomain come from. The domain is our input, our Codomain is what we can output.

Relations
- If $A$ and $B$ are sets, a relation $R$ between $A$ and $B$ is a subset of $S \subseteq A \times B$ . If $a \in A, b \in B$ , we say $a R b$ ( $a$ is related to $b$ ) if $(a, b) \in S$ .
- $R$ is:
  - left-total if $\forall a \in A, \exists b \in B : a R b$
  - functional if $\forall a \in A, c \in B, a R b, a R c ⟹ b = c$ .
If $A$ and $B$ are two sets, a function $f$ from $A \to B$ is a left-total, functional binary relation on $A, B$ .
Then $A$ is the Domain of $f$ and $B$ is the Codomain (indicates the output of where $f$ lives) of $f$
Example:
- $A = {2, 4, 6}$
- $B = {♡, ♢}$
- $S = {(2, ♡), (4, ♡), (6, ♢)}$
- Left-totality means a function is defined for every element in the domain.
- Functionality means that it passes the vertical line test.
Surjective
- If you have something in the codomain that's unmapped to, it's not surjective.
- MAT102 Term Test 1 Prep#Composition Proof Surjective