Set Operations

We combine sets using operations like union and intersection. Mastering these rules is essential for the Principle of Inclusion-Exclusion.

Basic Operations

Given two sets $A$ and $B$ within a universal set $U$:

1. Union ($A\cup B$): The set of elements in $A$ OR $B$ (or both).

• $x\in A\cup B⟺x\in A\vee x\in B$.

2. Intersection ($A\cap B$): The set of elements in BOTH $A$ and $B$.

• $x\in A\cap B⟺x\in A\wedge x\in B$.

3. Difference ($A\setminus B$ or $A-B$): Elements in $A$ but NOT in $B$.

1. Complement ($A^c$ or $\overline{A}$): Elements in $U$ but not in $A$.

De Morgan's Laws

These laws describe how complements interact with unions and intersections. They are crucial for "complementary counting" in complex scenarios.

Distributive Laws

Sets behave somewhat like algebra:

• $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
• $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

Cardinality Rules

• $|A\cup B|=|A|+|B|-|A\cap B|$
• $|A\setminus B|=|A|-|A\cap B|$

Practice Problems