Page 85 - Computer Science V2.0 Class 11

+ and ● are also called disjunction and conjunction, respectively.

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.

3.3.2 Important Theorems

Below we give some useful properties of Boolean algebra that can be proved using the above-mentioned properties.
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.
Theorem 2: Universal bounds: For ∀ a ∈ S,
(i) a + 1 = 1
(ii) a ● 0 = 0

We can verify Theorem 2 using Table 3.5

a a+ 1 a a.0

0 1 0 0
1 1 1 0
a+ 1 = 1 a.0 = 0

Table 3.5: a + 1 = 1 and a.0 = 0
Theorem 3: Absorption Law: ∀ a, b ∈ S

(i) a + a ● b = a
(ii) a ● (a + b) = a

Boolean Logic 71