The document defines a Boolean algebra as a set with binary operations of sum and product that satisfy closure, commutative, associative, distributive, identity, and complement laws. Examples of Boolean algebras include sets of binary values {1,0} and sets closed under union, intersection, and complement. Theorems proved include idempotent, involution, De Morgan's, and order properties. Boolean switching circuits can be described using series and parallel combinations of switches and satisfy the algebra of a Boolean algebra.

1. BOOLEAN ALGEBRA AND SETS
DEFINITION 1: A Boolean algebra is a set B with two binary operations sum (+, ∨, or ∪) and product (∗, ∧, or ∩)
such that:
B0. Closure Law
For any 𝑎, 𝑏 ⋲B, the sum 𝑎 + 𝑏 and the product 𝑎 ∗ 𝑏 exist and are unique elements in B.
B1. Commutative Law
𝑎 + 𝑏 = 𝑏 + 𝑎 and 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎
B2. Associative Law
𝑎 + ( 𝑏 + 𝑐) = ( 𝑎 + 𝑏) + 𝑐 and 𝑎 ∗ ( 𝑏 ∗ 𝑐) = ( 𝑎 ∗ 𝑏) ∗ 𝑐
B3. Distributive Law
𝑎 + ( 𝑏 ∗ 𝑐) = ( 𝑎 + 𝑏) ∗ ( 𝑎 + 𝑐) and 𝑎 ∗ ( 𝑏 + 𝑐) = ( 𝑎 ∗ 𝑏) + (𝑎 ∗ 𝑐)
B4. Identity
An additive identity 0 and multiplicative identity 𝑈 exist such that for any 𝑎 ⋲ 𝐵, 𝑎 + 0 = 𝑎 and 𝑎 ∗ 𝑈 = 𝑎
B5. Complement
For any 𝑎 ⋲ 𝐵there exists 𝑎′ ⋲ 𝐵 called the complement of 𝑎 such that 𝑎 + 𝑎′ = 𝑈 and 𝑎 ∗ 𝑎′ = 0.
Examples:
1. Let B = {1, 0} and let two operations + and ∗ be defined as follows:
+ 1 0 ∗ 1 0
1 1 1 1 1 0
0 1 0 0 0 0
Then B, or more precisely the triplet (B, +, ∗), is a Boolean algebra.
2. Let 𝒜 be a family of sets that is closed under operation of union, intersection, and complement. Then ( 𝒜,∪,∩) is
a Boolean algebra.
3. Let 𝓑 the set of propositions generated by variables 𝑝, 𝑞,….Then (𝓑, ∨, ∧) is a Boolean algebra.
Exercise:
Show that Example 2 is a Boolean Algebra using sets A = {a, b, c} and set B = {d, e}.
DUALITY IN BOOLEAN ALGEBRA
By definition, the dual of any statement in a Boolean algebra (B, +, ∗) is the statement that is derived by
interchanging + and ∗, and their identity elements U and 0, in the original statement; for example, the dual of
(𝑈 + 𝑎) ∗ ( 𝑏 + 0) = 𝑏 is (0 ∗ 𝑎) + ( 𝑏 ∗ 𝑈) = 𝑏.
Principle of Duality: The dual of any theorem in a Boolean algebra is also a theorem.

2. BASIC THEOREMS
THEOREM 1 (Idempotent Law): (𝑖) 𝑎 + 0 = 𝑎 (𝑖𝑖) 𝑎 ∗ 𝑈 = 𝑎.
THEOREM 2: (𝑖)A ∪ A’ = U (𝑖𝑖)A ∩ A’ = ⌀.
THEOREM 3 (Involution Law): ( 𝑎′)′ = 𝑎
THEOREM 4: (𝑖) 𝑈′ = 0 (𝑖𝑖)0′ = 𝑈
THEOREM 5 (DeMorgan’s Law): (𝑖)( 𝑎 + 𝑏)′ = 𝑎′ ∗ 𝑏′(𝑖𝑖)( 𝑎 ∗ 𝑏)′ = 𝑎′ + 𝑏′
Exercise:
1. Prove Theorem 2: (𝑖) 𝑎 + 𝑈 = 𝑈 (𝑖𝑖) 𝑎 ∗ 0 = 0
2. Prove Theorem 4: (𝑖) 𝑈′ = 0 (𝑖𝑖) 0′ = 𝑈
ORDER IN A BOOLEAN ALGEBRA
THEOREM 6: Let 𝑎, 𝑏 ⋲ 𝐵, a Boolean algebra. Then the following conditions are equivalent:
(1) 𝑎 ∗ 𝑏′ = 0 (2)𝑎 + 𝑏 = 𝑏 (3)𝑎′ + 𝑏 = 𝑈 (4)𝑎 ∗ 𝑏 = 𝑎
DEFINITION 2: Let 𝑎, 𝑏 ⋲ 𝐵, a Boolean algebra. Then 𝑎 𝑝𝑟𝑒𝑐𝑒𝑑𝑒 𝑏, denoted by 𝑎 ≲ 𝑏, if one of the properties in
Theorem 6 holds.
Examples:
1. Consider a Boolean algebra of sets ( 𝒜,∪,∩). Then A precedes B means that A⊂B. In other words, Theorem 6
states that if A is a subset of B, then the following conditions hold:
(1) A ∩ B’ = ⌀ (2) A ∪ B = B (3) A’ ∪ B = U (4) A ∩ B = A
2. Consider a Boolean algebra of propositions (𝓑, ∨, ∧). Then p precedes q means that p logically implies q.
THEOREM 7: The relation in a Boolean algebra B defined by 𝑎 ≲ 𝑏 is a partial order in B.
(1) 𝑎 ≲ 𝑎 for every 𝑎 ⋲ 𝐵 (Reflexive Law)
(2) 𝑎 ≲ 𝑏 and 𝑏 ≲ 𝑎 implies 𝑎 = 𝑏 (Anti-symmetric Law)
(3) 𝑎 ≲ 𝑏 and 𝑏 ≲ 𝑐 implies 𝑎 ≲ 𝑐 (Transitive Law)
Exercise:
Prove Theorem 7.

3. SWITCHING CIRCUITS DESIGNS
Let A, B, … denote electrical switches, and let A and A’ denote switches with the property that if one is on then
the other is off, and vice versa. Two switches, say A and B, can be connected by wire in a series or parallel
combination as follows:
Series combination, A ∧ B Parallel combination, A ∨ B
A Boolean switching circuit design means an arrangement of wires and switches that can be constructed by
repeated use of series and parallel combinations; hence it can be described by the use of connectives ∧ and ∨.
Circuit (1) A ∧ (B ∨ A’) Circuit (2) (A ∧ B’) ∨ [(A ∨ C) ∧ B]
Now let 1 and 0 denote respectively, that a switch or a circuit is on and that a switch or circuit is off.
*The tables describe the behavior of a series circuit A ∧ B, parallel circuit A ∨ B, and the relationship between
switch A and switch A’.
THEOREM 8: The algebra of Boolean switching circuits is a Boolean algebra.
Exercise:
1. Using circuit (1) in the example, when will the circuit be on and when will the circuit be off?
2. Construct s circuit for (A ∨ B) ∧ C ∧ (A’ ∨ B’ ∨ C’)
Reference:
Lipschuts, S. (2005). Schaums outline of theory and problem of set theory and related topics. New York: McGraw-
Hill Publishing Inc.
Raymund T. de la Cruz
MAEd – Mathematics
A B A ∧ B A B A ∨ B A A'
1 1 1 1 1 1 1 0
1 0 0 1 0 1 0 1
0 1 0 0 1 1
0 0 0 0 0 0
A B
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B
A
B
A’
C
A
B
A’
B’