Set Distributive Law

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A Set Distributive Law is an equality relation involving the union, [math]\displaystyle{ \cup }[/math] and intersection, [math]\displaystyle{ \cap }[/math] set operations.

[math]\displaystyle{ (B \cap C) = \{1,\;2,\;6 \},\quad (A \cap B)=\{1,\;2,\;3 \},\quad (A \cap C)=\{1,\;2,\;4 \} }[/math].
[math]\displaystyle{ (A \cup B) = \{1,\; 2,\; 3,\; 4,\; 5,\;6.\;7\},\quad (A \cup C) = \{1,\; 2,\; 3,\; 4,\; 5,\;6,\;8\} ,\quad (B \cup C) = \{1,\; 2,\; 3,\; 4,\;6,\;7, \;8\} }[/math]
(a) distributive property over set union: [math]\displaystyle{ A \cup (B \cap C) = \{1,\; 2,\; 3,\; 4,\; 5,\;6 \} \iff (A \cup B) \cap (A \cup C) = \{1,\; 2,\; 3,\; 4,\; 5,\;6 \} }[/math]
(b) distributive property over set intersection: [math]\displaystyle{ A \cap (B \cup C)= \{1,\; 2,\; 3,\; 4\} \iff (A \cap B) \cup (A \cap C) = \{1,\; 2,\; 3,\; 4\} }[/math]



Associative laws:
  • [math]\displaystyle{ (A \cup B) \cup C = A \cup (B \cup C)\,\! }[/math]
  • [math]\displaystyle{ (A \cap B) \cap C = A \cap (B \cap C)\,\! }[/math]
Distributive laws:
  • [math]\displaystyle{ A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\,\! }[/math]
  • [math]\displaystyle{ A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\,\! }[/math]
The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over union. However, unlike addition and multiplication, union also distributes over intersection.

Two additional pairs of laws involve the special sets called the empty set Ø and the universal set [math]\displaystyle{ U }[/math] ; together with the complement operator (AC denotes the complement of A). The empty set has no members, and the universal set has all possible members (in a particular context).

Identity laws:
  • [math]\displaystyle{ A \cup \varnothing = A\,\! }[/math]
  • [math]\displaystyle{ A \cap U = A\,\! }[/math]
Complement laws:
  • [math]\displaystyle{ A \cup A^C = U\,\! }[/math]
  • [math]\displaystyle{ A \cap A^C = \varnothing\,\! }[/math]
The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity elements for union and intersection, respectively(...)