Set Theory | Notion

Let $A$ and $B$ be sets

$x \in A$ means $x$ is an element of $A$
$x \not \in A$ means $x$ is not an element of $A$
$A \subseteq B$ ($A$ is a subset of $B$) means every element of $A$ is an element of $B$
$A = B$ means $A \subseteq B$ and $B \subseteq A$
$A \subsetneq B$ ($A$ is a proper subset of $B$) means $A \subseteq B$ but $A \neq B$
$\emptyset$ is the empty set, and $\emptyset \subseteq A$ for any set $A$
in set theory, $\{a, a, b\} = \{a, b\}$

$A \cup B$ is the union of $A$ and $B$
$A \cap B$ is the intersection of $A$ and $B$
$A \Delta B$ is the symmetric difference of $A$ and $B$
- in other notation
  
  $$ \{x : (x \in A \land x \not \in B) \lor (x \in B \land x \not \in A)\} $$
- Example: $\{1, 2, 3, 4\} \Delta \{3, 4, 5, 6\} = \{1, 2, 5, 6\}$
- $A \Delta B = B \Delta A = (A \cup B) - (A \cap B) = (A - B) \cup (B - A)$
$A$ and $B$ are disjoint if $A \cap B = \emptyset$
$\bar A$ is the complement of $A$
$A - B = \{x: x \in A \land x \not \in B\}$
- $\{1, 2, 3, 4\} - \{3, 4, 5, 6\} = \{1, 2\}$
- In general, $A - B \neq B - A$
$\bigcup_{i \in I} A_i = \{x : x \in A_i \text{ for some } i \in I\}$
- for example, if $I = \mathbb{N}$, then
$$ \bigcup_{i \in I} A_i = A_0 \cup A_1 \cup .... $$
$\bigcap_{i \in I} A_i = \{x : x \in A_i \text{ for all } i \in I\}$
- Demorgan’s Law :
$$ \overline{\bigcup_{i \in I}} A_i = \bigcap_{i \in I} \overline{A_i} $$

$$ \overline{\bigcap_{i \in I}} A_i = \bigcup_{i \in I} \overline{A_i} $$