Review of Set Theory and Functions

For sets:

Sets are a collection of elements
We write $x \in A$ if $x$ is an element of $A$
Union: $A \cup B = {x : x \in A \lor x \in B}$
Intersection: $A \cap B = {x : x \in A \land x \in B}$
Empty Set: $\emptyset$ is the set with no elements: $\emptyset = {}$
Two sets are distoint if $A \cap B = \emptyset$
Subset: $A \subseteq B$ if $x \in A$ implies $x \in B$
Set equivalence: both $A \subseteq B$ and $A \supseteq B$ is the same as $A = B$
$⋃_{i = 1}^{n} A_{i} = A_{1} \cup \dots \cup A_{n}$ and similar for intersection. If $n \to \infty$ then it equals $A_{1} \cup A_{2} \cup \dots$
- $x \in ⋃_{i = 1}^{\infty} A_{i}$ implies that there is some $A_{i}$ where $x \in A_{i}$
- $x \in ⋂_{i = 1}^{\infty} A_{i}$ implies that there for all $A_{i}$ then $x \in A_{i}$
Complement: $A^{c} = {x \in U : x \notin A}$ where $U$ is the universe of discourse. For our sakes usually $U = R$
De Morgan's Laws:
- $(A \cap B)^{c} = A^{c} \cup B^{c}$
- $(A \cup B)^{c} = A^{c} \cap B^{c}$

For functions:

Denote a function $f : A \to B$ as a function that maps elements from $A$ to elements in $B$ .
If $x \in A$ then $f (x) \in B$ is some element associated with $x$
$range (f) = {y \in B : \exists x \in A, y = f (x)}$ . This is also known as the image.
$domain (f) = A$