Chapter 4: Operations on Sets

4.1 Intersection of Sets

Definition of Intersection of Sets

The intersection of set $A$ and set $B$, written as $A \cap B$, is the set containing all elements that belong to both set $A$ and set $B$.

• Set notation: $A \cap B = \{x : x \in A \text{ and } x \in B\}$
• In a Venn diagram, $A \cap B$ represents the overlapping region shared by both sets.

Complement of the Intersection of Sets

The complement of the intersection of set $A$ and set $B$, written as $(A \cap B)'$, is the set of all elements in the universal set ($\xi$) that are not in $A \cap B$.

• Notation: $(A \cap B)' = \{x : x \in \xi \text{ and } x \notin (A \cap B)\}$

Solving Problems Involving Intersection of Sets

Problem-solving usually involves representing given information using Venn diagrams, establishing equations based on region cardinalities, and solving for unknown values.

4.2 Union of Sets

Definition of Union of Sets

The union of set $A$ and set $B$, written as $A \cup B$, is the set containing all elements that belong to set $A$, set $B$, or both.

• Set notation: $A \cup B = \{x : x \in A \text{ or } x \in B\}$
• In a Venn diagram, $A \cup B$ includes the entire area covered by set $A$ and set $B$ combined.

Complement of the Union of Sets

The complement of the union of set $A$ and set $B$, written as $(A \cup B)'$, is the set of all elements in the universal set ($\xi$) that do not belong to set $A$ or set $B$.

• Notation: $(A \cup B)' = \{x : x \in \xi \text{ and } x \notin (A \cup B)\}$
• In a Venn diagram, it is the region outside both circles $A$ and $B$.

4.3 Combined Operations on Sets

Combined Operations: Intersection and Union

Combined operations involve expressions with both $\cap$ and $\cup$, such as $(A \cup B) \cap C$ or $(A \cap B) \cup C$.

Order of Operations for Sets

1. Perform operations inside brackets $(\quad)$ first.
2. If no brackets are present, evaluate operations from left to right.

Relationship between Cardinality of Sets

For any two finite sets $A$ and $B$:

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

For three combined sets $A$, $B$, and $C$, Venn diagrams are divided into 8 distinct regions to analyze and count element frequencies systematically.