The Right Distributive Laws hold for all Boolean Algebras

The Right Distributive Laws of Boolean Algebra follow immediately from the Left Distributive Laws and the Commutative Laws. We can therefore add these laws directly to our table of:

Properties of Boolean Algebras
If (A, ∪, ∩, ¬, E∩, E∪ ) is a boolean algebra, then for every a, b, and c in A, the following identities hold. ----------------------------------------------------------------------------------------------------------------------------
a ∪ (b ∪ c) = (a ∪ b) ∪ c		a ∩ (b ∩ c) = (a ∩ b) ∩ c		Associativity
a ∪ (a ∩ b) = a		a ∩ (a ∪ b) = a		Absorption
a ∪ ¬a = E∩		a ∩ ¬a = E∪		Complements
a ∪ E∪= a		a ∩ E∩= a		Right identities
a ∪ b = b ∪ a		a ∩ b = b ∩ a		Commutativity
E∪ ∪ a = a		E∩ ∩ a = a		Left identities
a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c)		a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c)		Left Distributive Laws
(a ∩ b) ∪ c = (a ∪ c) ∩ (b ∪ c)		(a ∪ b) ∩ c = (a ∩ c) ∪ (b ∩ c)		Right Distributive Laws
---------------------------------------------------------------------------------------------------------------------------- Properties in black font are axioms.

The Binary Operations of Boolean Algebras together with their Identity Elements are Monoids

A monoid (M, ∗, e) is a semigroup (M, ∗) with a two-sided identity element e. Semigroups are associative magmas and a magma (M, ∗) is just a binary operation ∗: M×M→M on M.

Given a Boolean Algebra (A, ∩, ∪, ¬, E_∪, E_∩), we want to show that (A, ∩, E_∩) and (A, ∪, E_∪) are monoids. We start by noting that ∩: A×A→A and ∪: A×A→A are binary operations. So, (A, ∩) and (A, ∪) are, by definition, magmas. Next, we note that ∩ and ∪ both satisfy the associative law. So, (A, ∩) and (A, ∪) are, by definition, semigroups.

Next, recall that the absorption and compliment axioms imply that E_∪ and E_∩ are right identities of ∪ and ∩ respectively. We can use this fact together with the commutative laws to prove that (A, ∩, E_∩) and (A, ∪, E_∪) are monoids. The proof follows.

Poof that the Right Identities of ∪ and ∩ are also Left Identities
Let a be an element of A:

1		a ∪ E∪ = a		a ∩ E∩ = a		Since E∪ and E∩ are right identities.
2		E∪ ∪ a = a		E∩ ∩ a = a		Using the Commutative Laws.
Therefore, E∪ and E∩ are Left Identities of ∪ and ∩, respectively.
Therefore, E∪ and E∩ are two-sided identities of ∪ and ∩, respectively.
Therefore, (A, ∪, E∪ ) and (A, ∩, E∩) are monoids.

We can now add left identities to our list of properties of Boolean Algebras.

Properties of Boolean Algebras

a ∪ (b ∪ c) = (a ∪ b) ∪ c		a ∩ (b ∩ c) = (a ∩ b) ∩ c		Associativity
a ∪ (a ∩ b) = a		a ∩ (a ∪ b) = a		Absorption
a ∪ ¬a = E ∩		a ∩ ¬a = E ∪		Complements
a ∪ E ∪ = a		a ∩ E ∩ = a		Right identities
a ∪ b = b ∪ a		a ∩ b = b ∩ a		Commutativity
E∪ ∪ a = a		E∩ ∩ a = a		Left identities
a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c)		a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c)		Distributivity
Properties in blue font are axioms. Properties in black font are derived from the axioms.

The Distinguished Elements of Boolean Algebra are Identities

The absorption and compliment axioms of Boolean Algebra imply that the distinguished elements e_∩ and e_∪ are right identities of the conjunction and disjunction operations. Here are poofs: Let (A, ∩, ∪, ¬, e_∩, e_∪) be a Boolean algebra. Let a be an element of A. Let b = ¬a and define E_∩ = e_∪ and E_∪ = e_∩.

Proof that E∪ is a right identity of the disjunction operation ∪
a ∪ (a ∩ b) = a	Absorption axiom.	1
b = ¬a	By definition.	2
a ∪ (a ∩ ¬a) = a	By 1 and 2.	3
a ∩ ¬a = e∩	Complement axiom.	4
a ∪ e∩ = a	By 3 and 4.	5
E∪ = e∩	By definition.	6
a ∪ E∪ = a	By 5 and 6.	7
Therefore, E∪ is a right identity of the disjunction operation ∪.

Proof that E∩ is a right identity of the conjunction operation ∩
a ∩ (a ∪ b) = a	Absorption axiom.	1
b = ¬a	By definition.	2
a ∩ (a ∪ ¬a) = a	By 1 and 2.	3
a ∪ ¬a = e∪	Complement axiom.	4
a ∩ e∪ = a	By 3 and 4.	5
E∩ = e∪	By definition.	6
a ∩ E∩ = a	By 5 and 6.	7
Therefore, E∩ is a right identity of the conjunction operation ∩.

We can now add the existence of right identities to our list of properties of Boolean Algebras.

A Boolean Algebra is a 6-tuple (A, ∩, ∪, ¬, E∪, E∩) that satisfies the following Properties For all a, b, and c in A:
a ∪ (a ∩ b) = a	a ∩ (a ∪ b) = a	Absorption
a ∪ ¬a = E∩	a ∩ ¬a = E∪	Complements
a ∪ E∪ = a	a ∩ E∩ = a	Right identities
a ∪ (b ∪ c) = (a ∪ b) ∪ c	a ∩ (b ∩ c) = (a ∩ b) ∩ c	Associativity
a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c)	a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c)	Distributivity
a ∪ b = b ∪ a	a ∩ b = b ∩ a	Commutativity
Axioms are in blue font. Derived properties, just one of them so far—the existence of right identities—are listed in black font.

Notice that I have replaced e_∩ and e_∪ with E_∪ and E_∩ respectively. The ¬ operator is not a true analog to negation. Since a ∪ ¬a = E_∩ (the identity of ∩) rather than E_∪ (the identity of ∪), ¬a is not an inverse of a with respect to ∪. Likewise, ¬a is not and inverse of a with respect to ∩.

How to Read These Posts

Which Browser to Use

Some browsers, such as Internet Explorer, don't display math symbols correctly. This symbol ---> ∪ <--- should look like a U. If it looks like a square, it is being displayed incorrectly. Try switching to Firefox. It is available free on the Internet.

How to Adjust Font Size

If you are using Firefox and the table titled The Standard Axioms of Boolean Algebra in my previous post is unreadable, try adjusting the font size. Notice the 'ctrl' key on the bottom left of your keyboard. Now notice the '+' key next to the 'backspace' key at the top right of your keyboard. Press and hold the 'ctrl' key while tapping on the '+' key. This should make the font size grow larger. Watch what happens to the distributivity axioms in the table as you do this. They become squished together and unreadable.

Now notice the '-' key next to the '+' key. Press and hold the 'ctrl' key while tapping on the '-' key. The font size will shrink until the distributivity axioms separate and the table becomes readable.

Notation

The notation and concepts that I am using here is based on Symbolic Logic (5^th Edition) by Irving M. Copi. Pay particular attention to section 7.2: Axioms for Class Algebra.

Axioms

Although I am using Copi's notation, I have chosen the Axioms of Boolean Algebra given in Wikipedia as the “standard” axioms. These axioms more closely resemble the axioms of group theory and have, I suppose, a better claim to the designation “standard”.

The Standard Axioms of Boolean Algebra

A Boolean algebra is a 6-tuple consisting of a set A, equipped with two binary operations, a unary operation, and two distinguished elements. The first binary operation ∩: A × A → A is referred to as “conjunction”, “meet”, or “and”. The second binary operation ∪: A × A → A is referred to as “disjunction”, “join”, or “or”. The unary operation ¬: A → A is referred to as “compliment” or “not”. The distinguished elements are often denoted as 0 and 1. However, I will denote these elements as e_∩ and e_∪.

The 6-tuple (A, ∩, ∪, ¬, e_∩, e_∪) is called a “Boolean algebra” if it satisfies certain stated axioms. The axioms differ depending upon the needs of a particular author. The five following axiom pairs are a particularly common formulation of Boolean algebra. For all a, b, and c in A:

The Standard Axioms of Boolean Algebra
a ∪ (a ∩ b) = a	a ∩ (a ∪ b) = a	absorption
a ∪ (b ∪ c) = (a ∪ b) ∪ c	a ∩ (b ∩ c) = (a ∩ b) ∩ c	associativity
a ∪ ¬a = e∪	a ∩ ¬a = e∩	complements
a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c)	a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c)	distributivity
a ∪ b = b ∪ a	a ∩ b = b ∩ a	commutativity