Complement Law

A Complement Law is a Boolean Algebra law of complement sets.

• AKA: Complement Set Law.
• Context:
  • It can be defined as: Given a set [math]\displaystyle{ A }[/math] and its complement [math]\displaystyle{ A^C }[/math] then

1. [math]\displaystyle{ A\cup \overline{A}=U }[/math]

2. [math]\displaystyle{ A\cap \overline{A}=\emptyset }[/math]

where [math]\displaystyle{ U }[/math] is the universe set and [math]\displaystyle{ \emptyset }[/math] is the empty set, such that [math]\displaystyle{ U^C=\emptyset }[/math] and [math]\displaystyle{ \emptyset^C=U }[/math].

• It can be also be expressed as

1. [math]\displaystyle{ A+A^c=1 }[/math]

2. [math]\displaystyle{ A*A^c =0 }[/math]

• Example(s)
  • [math]\displaystyle{ (A^C)^C=A }[/math], Double complement Law.
• Counter-Example(s)
  • DeMorgan's law.
  • Commutative Laws
  • Associative Laws.
  • Distributive Laws
  • Identity Laws
  • Idempotent Laws
• See: DeMorgan's Laws, Set Theory, Absolute Complement, Involution (Mathematics), Proper Subset.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Complement_(set_theory)#Properties Retrieved:2017-6-4.
  • Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:

De Morgan's laws:

• [math]\displaystyle{ \left(A \cup B \right)^{\complement}=A^{\complement} \cap B^{\complement} . }[/math]
• [math]\displaystyle{ \left(A \cap B \right)^{\complement}=A^{\complement} \cup B^{\complement} . }[/math]

Complement laws:

• [math]\displaystyle{ A \cup A^{\complement} = U . }[/math]
• [math]\displaystyle{ A \cap A^{\complement} =\varnothing . }[/math]
• [math]\displaystyle{ \varnothing^{\complement} =U. }[/math]
• [math]\displaystyle{ U^{\complement} =\varnothing. }[/math]
• [math]\displaystyle{ \text{If }A\subset B\text{, then }B^{\complement}\subset A^{\complement}. }[/math]

  (this follows from the equivalence of a conditional with its contrapositive).

Involution or double complement law:

• [math]\displaystyle{ (A^{\complement})^{\complement}=A. }[/math]

Relationships between relative and absolute complements:

• [math]\displaystyle{ A \setminus B = A \cap B^\complement. }[/math]
• [math]\displaystyle{ (A \setminus B)^\complement = A^\complement \cup B. }[/math]

Relationship with set difference:

• [math]\displaystyle{ A^\complement \setminus B^\complement = B \setminus A. }[/math]

The first two complement laws above show that if A is a non-empty, proper subset of U, then {A, A^∁} is a partition of U.

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