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Boolean algebra, like any other axiomatic mathematical structure or algebraic system, can be characterized by specifying a number of fundamental things: 1. The domain of the algebra, that is, the set of elements over which the algebra is defined 2. A set of operations to be performed on the elements 3. A set of postulates, or axioms, accepted as premises without proof 4. A set of consequences called theorems, laws, or rules, which are deduced from the postulates
ĐẠI HỌC CÔNG NGHỆ VIETNAM NATIONAL UNIVERSITY, HANOI University of Engineering & Technology Digital Design Lecture 3: Boolean and Switching Algebra Xuan-Tu Tran, PhD Faculty of Electronics and Telecommunication Smart Integrated Systems (SIS) Laboratory Email: tutx@vnu.edu.vn www.uet.vnu.edu.vn/~tutx ĐẠI HỌC CÔNG NGHỆ History Developed by George Boole in his book (a treatise): “An Investigation of the Laws of Thought” no application was made of Boolean Algebra until the late 1930s Nakashima in Japan (1937) and Shannon at MIT (1938), each independently applied the algebra of Boole to the analysis of networks of relays (in telephone systems) 10/8/2010 Xuan-Tu Tran ĐẠI HỌC CÔNG NGHỆ The Huntington postulates In 1904, Huntington found that all of the results and implications of the algebra described by Boole could be derived from only six basic postulates Huntington postulates: The set {B, +, ·, ¯ } B is the set of elements or constants of the algebra the symbols + and · are two binary operators(*) the overbar ¯ is a unary operator(*) is a Boolean algebra if the following hold true: (*) 10/8/2010 The terms binary operator and unary operator refer to the number of arguments involved in the operation: two or one, respectively Xuan-Tu Tran ĐẠI HỌC CÔNG NGHỆ The Huntington postulates Closure (khép kín) For all elements a and b in the set B, a+b∈B a·b∈B Existing and elements There exists a element in B such that for every element a in B, + a = a + = a There exists a element in B such that for every element a in B, a = a · = a 10/8/2010 Xuan-Tu Tran ĐẠI HỌC CÔNG NGHỆ The Huntington postulates Commutativity (giao hoán) a+b=b+a a·b=b·a Distributivity (phân phối) a · (b + c) = a · b + a · c a + (b · c) = (a + b) · (a + c) 10/8/2010 Xuan-Tu Tran The Huntington postulates ĐẠI HỌC CÔNG NGHỆ ∀a ∈ B, there exists an element a-bar in the set B such that a + a =1 a⋅a = There exists at least two distinct elements in B Switching algebra is a Boolean algebra in which the number of elements in the set B is precisely 10/8/2010 Xuan-Tu Tran ĐẠI HỌC CÔNG NGHỆ Switching Algebra is a Boolean algebra in which the number of elements in the set B is precisely The two binary operators, represented by the signs + and ·, are called the OR and the AND, respectively The unary operator, represented by the overbar ¯ , is called the NOT or the complement operator 10/8/2010 Xuan-Tu Tran ĐẠI HỌC CÔNG NGHỆ Algebra Implications Theorem – idempotence (i) a+a = a (ii) a⋅a = a (Định lý hấp thụ) Proof a + a = (a + a ) ⋅1 = (a + a ) ⋅ (a + a ) = a + a⋅a = a+0 =a 10/8/2010 P-2(ii) P-5(i) P-4(ii) P-5(ii) P-2(i) Xuan-Tu Tran a ⋅ a= a ⋅ a + = a⋅a + a⋅a = a ⋅ (a + a ) = a ⋅1 =a P-2(i) P-5(ii) P-4(i) P-5(i) P-2(ii) ĐẠI HỌC CÔNG NGHỆ Algebra Implications Theorem (i) (ii) Proof a ⋅0 = + a⋅0 = a⋅a + a ⋅0 = a ⋅ ( a + 0) = a ⋅ (a ) =0 a ⋅0 = 0⋅a = a +1 = 1+ a = a +1 P-2(i) P-5(ii) Principle of duality P-4(i) P-2(ii) P-5(i) P-4(ii) P-2(i) P-2(ii) P-5(ii) P-5(i) Exercise for students 10/8/2010 Xuan-Tu Tran ĐẠI HỌC CÔNG NGHỆ Algebra Implications Theorem Let a be an element of B Then a-bar is unique Proof Assume that a has distinct complements (not equal), a-bar and b Then by P-5, we must have that: a +b =1 and a + a =1 and a⋅a = and a ⋅b = 10/8/2010 a = a ⋅1 b = b ⋅1 = a ⋅ ( a + b ) = b( a + a ) = a ⋅a + a ⋅b = b⋅a + b⋅a = + a ⋅b = 0+b⋅a = a ⋅b = a ⋅b Xuan-Tu Tran a =b P-2(ii) P-4(i) P-2(i) 10 ... elements in B Switching algebra is a Boolean algebra in which the number of elements in the set B is precisely 10/8/2010 Xuan-Tu Tran ĐẠI HỌC CÔNG NGHỆ Switching Algebra is a Boolean algebra in... NGHỆ Algebra Implications – Basic logic operations From this point, we will restrict our attention to switching algebras only Switching algebra is basically a two-element Boolean algebra. .. the algebra the symbols + and · are two binary operators(*) the overbar ¯ is a unary operator(*) is a Boolean algebra if the following hold true: (*) 10/8/2010 The terms binary operator and