Theorem 5.18: Algebra of Set Operations

Vocabulary flashcards covering the fundamental laws and equalities from Theorem 5.18 (Algebra of Set Operations).

Theorem 5.18 (Algebra of Set Operations)

A theorem stating that if $A$, $B$, and $C$ are subsets of a universal set $U$, then specific algebraic set equalities hold regarding the empty set, universal set, idempotent laws, commutative laws, associative laws, and distributive laws.

Properties of the Empty Set and the Universal Set

The set operational identities stating that $A \times \emptyset = \emptyset$, $A \cup \emptyset = A$, $A \cap U = A$, and $A \cup U = U$.

Idempotent Laws

The set equalities stating that operating on a set with itself yields the same set: $A \cap A = A$ and $A \cup A = A$.

Commutative Laws

The set equalities stating that the order of the sets does not change the result: $A \cap B = B \cap A$ and $A \cup B = B \cup A$.

Associative Laws

The set equalities stating that grouping does not affect the outcome of consecutive identical operations: $(A \cap B) \cap C = A \cap (B \cap C)$ and $(A \cup B) \cup C = A \cup (B \cup C)$.

Distributive Laws

The set equalities stating that intersection distributes over union, and union distributes over intersection: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ and $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

$$A \cup B =$$

$$B \cup A$$

$$A \cap B =$$

$$B \cap A$$

$$A \times \emptyset =$$

$$\emptyset$$

$$A \cup \emptyset =$$

$$A$$

$$A \cap U =$$

$$A$$

$$A \cup U =$$

$$U$$

$$A \cap A =$$

$$A$$

$$A \cup A =$$

$$A$$

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

$$A \cap (B \cap C)$$

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

$$A \cup (B \cup C)$$

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

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

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

$(A \cup B) \cap (A \cup C)$}],