2.3 De Morgan's laws and algebraic properties of sets

De Morgan's laws and algebraic properties of sets are key concepts in set theory. They provide rules for manipulating sets and their complements, helping us simplify complex set expressions.

These laws and properties form the foundation for solving set problems. Understanding them allows us to work with sets more efficiently, making it easier to analyze relationships between different sets and perform set operations.

De Morgan's Laws and Complement Laws

De Morgan's Laws for Set Operations

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• First law: The complement of the union of two sets $A$ and $B$ is equal to the intersection of their complements: $\overline{A \cup B} = \overline{A} \cap \overline{B}$
• Second law: The complement of the intersection of two sets $A$ and $B$ is equal to the union of their complements: $\overline{A \cap B} = \overline{A} \cup \overline{B}$
• Generalization: De Morgan's laws can be extended to any finite number of sets
• Example: Let $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$. Then $\overline{A \cup B} = \overline{\{1, 2, 3, 4\}} = \{5, 6, 7, ...\}$ and $\overline{A} \cap \overline{B} = \{5, 6, 7, ...\} \cap \{1, 5, 6, 7, ...\} = \{5, 6, 7, ...\}$

Complement Laws for Sets

• Law of double complement: The complement of the complement of a set $A$ is the set itself: $\overline{\overline{A}} = A$
• Law of the universe: The union of a set $A$ and its complement $\overline{A}$ is the universal set $[U](https://www.fiveableKeyTerm:u)$: $A \cup \overline{A} = U$
• Law of the empty set: The intersection of a set $A$ and its complement $\overline{A}$ is the empty set $\emptyset$: $A \cap \overline{A} = \emptyset$
• Example: Let $U = \{1, 2, 3, 4, 5\}$ and $A = \{1, 2, 3\}$. Then $\overline{A} = \{4, 5\}$, $\overline{\overline{A}} = \{1, 2, 3\}$, $A \cup \overline{A} = \{1, 2, 3, 4, 5\}$, and $A \cap \overline{A} = \emptyset$

Commutative, Associative, and Distributive Properties

Commutative Properties of Union and Intersection

• Union: The union of sets $A$ and $B$ is the same as the union of sets $B$ and $A$: $A \cup B = B \cup A$
• Intersection: The intersection of sets $A$ and $B$ is the same as the intersection of sets $B$ and $A$: $A \cap B = B \cap A$
• Order independence: The order of the sets in a union or intersection operation does not affect the result
• Example: Let $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$. Then $A \cup B = \{1, 2, 3, 4\} = B \cup A$ and $A \cap B = \{2, 3\} = B \cap A$

Associative Properties of Union and Intersection

• Union: The union of sets $A$, $B$, and $C$ can be performed in any order: $(A \cup B) \cup C = A \cup (B \cup C)$
• Intersection: The intersection of sets $A$, $B$, and $C$ can be performed in any order: $(A \cap B) \cap C = A \cap (B \cap C)$
• Grouping independence: The grouping of sets in a union or intersection operation does not affect the result
• Example: Let $A = \{1, 2\}$, $B = \{2, 3\}$, and $C = \{3, 4\}$. Then $(A \cup B) \cup C = \{1, 2, 3\} \cup \{3, 4\} = \{1, 2, 3, 4\} = A \cup (B \cup C)$ and $(A \cap B) \cap C = \{2\} \cap \{3, 4\} = \emptyset = A \cap (B \cap C)$

Distributive Properties of Union and Intersection

• Union over intersection: The union of a set $A$ with the intersection of sets $B$ and $C$ is equal to the intersection of the unions of $A$ with $B$ and $A$ with $C$: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
• Intersection over union: The intersection of a set $A$ with the union of sets $B$ and $C$ is equal to the union of the intersections of $A$ with $B$ and $A$ with $C$: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
• Distribution of operations: Union distributes over intersection and intersection distributes over union
• Example: Let $A = \{1, 2\}$, $B = \{2, 3\}$, and $C = \{3, 4\}$. Then $A \cup (B \cap C) = \{1, 2\} \cup \{3\} = \{1, 2, 3\} = (\{1, 2\} \cup \{2, 3\}) \cap (\{1, 2\} \cup \{3, 4\}) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = \{1, 2\} \cap \{2, 3, 4\} = \{2\} = (\{1, 2\} \cap \{2, 3\}) \cup (\{1, 2\} \cap \{3, 4\}) = (A \cap B) \cup (A \cap C)$

Identity, Idempotent, and Absorption Properties

Identity Properties of Union and Intersection

• Union with the empty set: The union of any set $A$ with the empty set $\emptyset$ is the set $A$ itself: $A \cup \emptyset = A$
• Intersection with the universal set: The intersection of any set $A$ with the universal set $U$ is the set $A$ itself: $A \cap U = A$
• Identity elements: The empty set is the identity element for union, and the universal set is the identity element for intersection
• Example: Let $A = \{1, 2, 3\}$, $\emptyset = \{\}$, and $U = \{1, 2, 3, 4, 5\}$. Then $A \cup \emptyset = \{1, 2, 3\} \cup \{\} = \{1, 2, 3\} = A$ and $A \cap U = \{1, 2, 3\} \cap \{1, 2, 3, 4, 5\} = \{1, 2, 3\} = A$

Idempotent Properties of Union and Intersection

• Union: The union of a set $A$ with itself is the set $A$: $A \cup A = A$
• Intersection: The intersection of a set $A$ with itself is the set $A$: $A \cap A = A$
• Redundancy: Performing a union or intersection operation on a set with itself does not change the set
• Example: Let $A = \{1, 2, 3\}$. Then $A \cup A = \{1, 2, 3\} \cup \{1, 2, 3\} = \{1, 2, 3\} = A$ and $A \cap A = \{1, 2, 3\} \cap \{1, 2, 3\} = \{1, 2, 3\} = A$

Absorption Properties of Union and Intersection

• Union: The union of a set $A$ with the intersection of $A$ and another set $B$ is the set $A$: $A \cup (A \cap B) = A$
• Intersection: The intersection of a set $A$ with the union of $A$ and another set $B$ is the set $A$: $A \cap (A \cup B) = A$
• Absorption of operations: A set absorbs the result of an intersection or union operation with itself and another set
• Example: Let $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$. Then $A \cup (A \cap B) = \{1, 2, 3\} \cup \{2, 3\} = \{1, 2, 3\} = A$ and $A \cap (A \cup B) = \{1, 2, 3\} \cap \{1, 2, 3, 4\} = \{1, 2, 3\} = A$