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Unformatted text preview: ECE 201 Digital Logic Chapter 3 1 of 36 3.1 Boolean Algebra For a mathematical system which consists of a set B of elements and operators, + and * , to be a Boolean Algebra , the following properties must hold. 1. Closure – The operators + and * are closed. That is, for all x, y ∈ B, then x + y ∈ B and x * y ∈ B 2. Identity Elements – For all x ∈ B , there exists identity elements for both + and * . e + x = x + e = x and e * x = x * e = x 3. Commutativity – The operations + and * are commutative . x + y = y + x and x * y = y * x 4. Distributivity – The operations + and * are distributive . x + (y * z) = (x + y)*(x + z) and x * (y + z) = (x * y)+(x * z) 5. Complement Elements – For all x ∈ B , there exists a complement of x, denoted X , x’ or x c , where. x + x’ = 1 and x * x’ = 0 Note: Though the notation X is used in this text and has been considered the standard notation for Boolean complements, x’ is now generally used simply because it can be more easily produced and stored in an ASCII world, and I, therefore, will use x’ in these notes and in class. 6. Elements – There are at least two elements x, y ∈ B, such that x ≠ y . The elements of the algebra are called constants and should not be confused with the binary numbers 0 and 1, though typically these symbols are used. A symbol which represents an element is called a variable . Note: Since the operations + and * are not defined, and the algebra can have more than two elements, we have actually described a class of algebras. ECE 201 Digital Logic Chapter 3 2 of 36 Principle of Duality – In a Boolean Algebra, if a statement is true for + and , then it is true for * and 1 (and vice versa). 3.2 Boolean Algebra Theorems Theorem 3.1 – The element x’ is uniquely determined by x . For this reason, the symbol ’ is considered a unary (monadic) operator for the algebra. Theorem 3.2 – For each element x ∈ B : ( a ) x + 1 = 1 ( b ) x . 0 = 0 Theorem 3.3 – Each of the identity elements in a Boolean is the complement of the other. ( a ) 0’ = 1 ( b ) 1’ = 0 Theorem 3.4 – Idempotent Law. For each element x ∈ B : ( a ) x + x = x ( b ) x . x = x Theorem 3.5 – Involution Law. For each element x ∈ B : (x’)’= x Theorem 3.6 – Absorbtion Law. For each element x, y ∈ B : ( a ) x + xy = x ( b ) x(x + y) = x Theorem 3.7 – For each element x, y ∈ B : ( a ) x + x’y = x + y ( b ) x(x’ + y) = xy Theorem 3.8 – Associativity. For each x, y, z ∈ B : ( a ) x +(y + z) = (x + y)+z ( b ) x(y z) = (x y)z Theorem 3.8 – De Morgan’s Law. For each x, y, z ∈ B : ( a ) (x + y)’ = x’y’ ( b ) (x y)’ = (x’ + y’) Precedence of Operators (1) Parentheses, (2) NOT , (3) AND , and (4) OR ECE 201 Digital Logic Chapter 3 3 of 36 3.3 TwoValued Boolean Algebras “ OR ” Operation “ AND ” Operation y y x + y 0 1 x + y 0 1 x 0 0 1 x 0 0 0 1 1 1 1 0 1 3.4 Boolean Functions Consider the function F = xy’ + xyz’ + y’z In the above function, x, y’, y, z’,...
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 Spring '08
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 Boolean Algebra, Logic gate

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