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Lec15_Boolean_Algebra - Discrete Structures TDS1191 Lecture...

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Discrete Structures 1 [Lecture 15][Boolean Algebra] Discrete Structures TDS1191 - Lecture 15 - Boolean Algebra
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Discrete Structures 2 [Lecture 15][Boolean Algebra] Boolean Algebra Invented by and named after George Boole. Commonly, and especially in computer science and digital electronics, this term is used to mean two-value logic . Provide the operations and rules for working with the set {0,1}. Map logical propositions to symbols. Permit manipulation of logic statements using Mathematics. Invented by and named after George Boole. Commonly, and especially in computer science and digital electronics, this term is used to mean two-value logic . Provide the operations and rules for working with the set {0,1}. Map logical propositions to symbols. Permit manipulation of logic statements using Mathematics. Definition: A Boolean algebra written as ( B , +, . , , 0, 1), consists of a set B containing distinct elements 0 and 1, binary operators + and . on B , and a unary operator on B , such that the identity laws, complement laws, associative laws, commutative laws and distributive laws holds. Definition: A Boolean algebra written as ( B , +, . , , 0, 1), consists of a set B containing distinct elements 0 and 1, binary operators + and . on B , and a unary operator on B , such that the identity laws, complement laws, associative laws, commutative laws and distributive laws holds.
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Discrete Structures 3 [Lecture 15][Boolean Algebra] Identity Law x + 0 = x , x . 1 = x Complement Law x + x = 1 x . x = 0 Associative Law ( x + y ) + z = x + ( y + z ) ( x . y ) . z = x . ( y . z ) Identity Law x + 0 = x , x . 1 = x Complement Law x + x = 1 x . x = 0 Associative Law ( x + y ) + z = x + ( y + z ) ( x . y ) . z = x . ( y . z ) Commutative Law x + y = y + x x . y = y . x Distributive Law x + ( y . z ) = ( x + y ) . ( x + z ) x . ( y + z ) = x . y + x . z Commutative Law x + y = y + x x . y = y . x Distributive Law x + ( y . z ) = ( x + y ) . ( x + z ) x . ( y + z ) = x . y + x . z
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Discrete Structures 4 [Lecture 15][Boolean Algebra] Note: 0 and 1 in the previous definition are merely symbolic names. In general, they have nothing to do with the numbers 0 and 1. + and are merely binary operators. In general, they have nothing to do with the ordinary addition and multiplication. Some examples of Boolean algebras are (a) ({ 0, 1 }, , , , 0, 1 ) (b) ( P(U) , , , f8e5 , , U ) where (i) and plays the role of +. (ii) and plays the role of . (iii) and f8e5 plays the role of complement. (iv) plays the role of 0 and the universal set U plays the role of 1. Note: 0 and 1 in the previous definition are merely symbolic names. In general, they have nothing to do with the numbers 0 and 1. + and are merely binary operators. In general, they have nothing to do with the ordinary addition and multiplication. Some examples of Boolean algebras are (a) ({ 0, 1 }, , , , 0, 1 ) (b) ( P(U) , , , f8e5 , , U ) where (i) and plays the role of +.
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