Page 92 - Computer Science V2.0 Class 11

Ø Basic properties of Boolean Algebra are:

1. Closure Property
(1) a + b ∈ S, ∀ a, b ∈ S
(2) a ● b ∈ S, ∀ a, b ∈ S
2. Commutative Property
(1) a + b = b + a, ∀ a, b ∈ S

(2) a ● b = b ● a, ∀ a, b ∈ S
3. Distributive Property
(1) a + (b ● c) = (a + b) ● (a + c), ∀ a, b, c ∈ S
(2) a ● (b + c) = a ● b + a ● c, ∀ a, b, c ∈ S

4. Identity Property
There exist two elements in S, denoted by 0 and 1, called identity of + and ● respectively, satisfying
(1) a + 0 = a, ∀ a ∈ S
(2) a ● 1 = a, ∀ a ∈ S
5. Complementarity Property

For each a ∈ S, there exists an element in S, denoted by a', such that
(1) a + a' = 1
(2) a ● a' = 0

a' is called the complement of a.
Ø Basic Theorems of Boolean Algebra are:
Ø Theorem 1: Uniqueness of complements: For each element a ∈ S, its complement is unique. The complement
of 0 is 1 and the complement of 1 is 0.
Ø Theorem 2: Universal bounds: For ∀ a ∈ S,
(i) a + 1 = 1
(ii) a ● 0 = 0

Ø Theorem 3: Absorption Law: ∀ a, b ∈ S
(i) a + a ● b = a

(ii) a ● (a + b) = a
Ø Theorem 4: Idempotent Law: ∀ a ∈ S
(i) a + a = a
(ii) a ● a = a
Ø Theorem 5: Involution: ∀ a ∈ S, (aꞌ)ꞌ = a
Ø Theorem 6: Associative Law: ∀, a, b, c ∈ S,
(i) (a + b) + c = a + (b + c)

(ii) (a ● b) ● c = a ● (b ● c)
Ø Theorem 7: De Morgan’s laws: ∀ a, b ∈ S
(i) (a + b)' = a' ● b'

(ii) (a ● b)' = a' + b'

78 Touchpad Computer Science (Ver. 2.0)-XI