Page 73 - Computer Science Class 11 With Functions
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+ 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







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