Set Distributive Law

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

• AKA: Set Theory Distributive Law, Set Union and Intersection Distributive Law.
  • It can be defined as:
    • [math]\displaystyle{ A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\,\! }[/math], (distributive property over set union)
    • [math]\displaystyle{ A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\,\! }[/math] (distributive property over set intersection)
  • It is analogous to the distributive laws involving the binary operations, addition (+) and multiplication (×), of numbers.
• Example(s):
  • Let's consider the following sets: [math]\displaystyle{ A = \{1,\; 2,\; 3,\; 4,\; 5\} }[/math], [math]\displaystyle{ B=\{1,\; 2,\; 3,\; 6,\; 7\} }[/math] and [math]\displaystyle{ C=\{ 1,\; 2,\; 4,\; 6,\;8\} }[/math]

[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]

Thus,

(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]

• Counter-Example(s)
  • Associative Set Law.
  • Commutative Set Law.
  • Complement Set Law.
  • Identity Set Law.
  • Double Complement Law.
  • De Morgan's Laws.
  • Idempotent Law.
• See: Distributive Operation Relation, Venn Diagram, Set Theory, Linear Logic, Binary Operation, Union (Set Theory), Intersection (Set Theory), Identity (Mathematics), Commutative Operation, Associativity, Distributivity, Empty Set, Universal Set, Complement (Set Theory), Identity Element.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Algebra_of_sets#The_fundamental_laws_of_set_algebra Retrieved:2017-6-25.
  • The binary operations of set union ([math]\displaystyle{ \cup }[/math] ) and intersection ([math]\displaystyle{ \cap }[/math] ) satisfy many identities. Several of these identities or "laws" have well established names.

     Commutative laws:

    • [math]\displaystyle{ A \cup B = B \cup A\,\! }[/math]
    • [math]\displaystyle{ A \cap B = B \cap A\,\! }[/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 (A^C 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(...)

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