Page 80 - Computer Science Class 11 Without Functions
P. 80

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, (a’)’ = 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’




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