Set Theory
Notations, Symbols & Formulas
1. Symbols & Notations
| Symbol | Meaning | Example |
|---|---|---|
| $$ \in $$ | Belongs to (Element is inside set) |
$$ 3 \in \{1, 2, 3\} $$ |
| $$ \notin $$ | Does not belong to | $$ 4 \notin \{1, 2, 3\} $$ |
| $$ \subset $$ | Proper Subset (A is inside B) |
$$ \{1\} \subset \{1, 2\} $$ |
| $$ \phi $$ or $$ \{\} $$ | Null / Empty Set (Set with no elements) |
$$ A = \phi $$ |
| $$ U $$ or $$ \xi $$ | Universal Set (Contains all sets) |
$$ U = \{1, 2, 3…\} $$ |
2. Operations on Sets
Union ($$ A \cup B $$)
Elements in A OR B (All combined).
$$ \{x : x \in A \text{ or } x \in B\} $$
Intersection ($$ A \cap B $$)
Elements in A AND B (Common part).
$$ \{x : x \in A \text{ and } x \in B\} $$
Difference ($$ A – B $$)
Elements in A but NOT in B.
$$ \{x : x \in A, x \notin B\} $$
3. Important Formulas & De Morgan’s Law
Cardinal Number Formula:
For any two finite sets A and B:
$$ n(A \cup B) = n(A) + n(B) – n(A \cap B) $$
De Morgan’s Law 1:
$$ (A \cup B)’ = A’ \cap B’ $$
De Morgan’s Law 2:
$$ (A \cap B)’ = A’ \cup B’ $$
*Note: $$ A’ $$ or $$ A^c $$ denotes the Complement of set A (Everything in U that is not in A).