Set Operations and Venn Diagrams

Venn Diagrams

Venn diagrams are visual tools used to represent relationships between sets. The universal set (U) is typically represented by a rectangle, and its subsets are represented by closed curves, usually circles.

Operations on Sets

1. Union of Sets ($\cup$) The union of two sets A and B is the set of all elements that are in set A, or in set B, or in both. Common elements are taken only once.

$$A \cup B = \{x : x \in A \text{ or } x \in B\}$$

2. Intersection of Sets ($\cap$) The intersection of two sets A and B is the set of all elements which are common to both A and B.

$$A \cap B = \{x : x \in A \text{ and } x \in B\}$$

• Disjoint Sets: If two sets have no elements in common ($A \cap B = \phi$), they are called disjoint sets.

3. Difference of Sets (-) The difference of sets A and B (in that order) is the set of elements which belong to A but not to B.

$$A - B = \{x : x \in A \text{ and } x \notin B\}$$

• Note that in general, $A - B \ne B - A$.

4. Complement of a Set (') Let U be the universal set. The complement of a set A, denoted by $A'$, is the set of all elements of U which are not in A.

$$A' = \{x : x \in U \text{ and } x \notin A\} = U - A$$

Properties & Laws of Set Operations

• Commutative Law: $A \cup B = B \cup A$; $A \cap B = B \cap A$
• Associative Law: $(A \cup B) \cup C = A \cup (B \cup C)$; $(A \cap B) \cap C = A \cap (B \cap C)$
• Distributive Law: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
• De Morgan's Laws: These are crucial for logic and set theory.

$$(A \cup B)' = A' \cap B'$$ $$(A \cap B)' = A' \cup B'$$

Example 1: Union and Intersection

Question: Let $A = \{2, 4, 6, 8\}$ and $B = \{6, 8, 10, 12\}$. Find $A \cup B$ and $A \cap B$.

Solution:

• Union: We list all unique elements from both sets. $A \cup B = \{2, 4, 6, 8, 10, 12\}$
• Intersection: We list only the elements that are common to both sets. $A \cap B = \{6, 8\}$

Example 2: Difference and Complement

Question: Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{1, 2, 3, 4\}$, and $B = \{2, 4, 6, 8\}$. Find $A - B$, $B - A$, and $A'$.

Solution:

• Difference (A - B): Elements in A but not in B. $A - B = \{1, 3\}$
• Difference (B - A): Elements in B but not in A. $B - A = \{6, 8\}$
• Complement (A'): Elements in U but not in A. $A' = \{5, 6, 7, 8, 9\}$