Distributive Properties in Set Theory

\[A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\] \[A \cap (B \cup C) = (A \cap B) \cup (A \cap C).\]

Examples

Let set $A = \{1, 2\}$, set $B = \{2, 3\}$, and set $C=\{2, 4\}$.

Union
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C) = \{1, 2\}$
Intersection
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C) = \{2\}$

Logical Equivalences

\[p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)\] \[p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r).\]

Examples

  • $p$: True (T).
  • $q$: False (F).
  • $r$: True (T).
Conjunction
T or (F and T) $=$ T, which is logically equivalent to: (T or F) and (T or T) $=$ T.
Disjunction
T and (F or T) = T, which is logically equivalent to: (T and F) or (T and T) = T.

Meaning of the used symbols

Symbol Meaning
$ \{ \} $ set
$ \cup $ union of sets
$ \cap $ intersection of sets

Relation to other Axioms

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