Lecture 2- Boolean

Lecture 2- Boolean - ELEC151 Spring 2011 L.Yobas Lecture 2...

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ELEC151 Spring 2011 – L. Yobas Lecture 2 – 1 Lecture 2 Boolean Algebra & Logic Gates ELEC151 Digital Circuits and Systems Spring 2011 Instructor: Levent Yobas
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ELEC151 Spring 2011 – L. Yobas Lecture 2 – 2 Lecture Overview Boolean Algebra Boolean Functions Logic Gates Implementing Boolean Functions using Logic Gates Canonical and Standard Forms Inside logic gates TTL and CMOS logic Electrical characteristics: Signal levels, Noise margins, Fan- out, Speed Reading assignment: Chapter 2.1 to 2.8
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ELEC151 Spring 2011 – L. Yobas Lecture 2 – 3 Logic function Digital system works on binary number which has only two elements, ‘0’, and ‘1’ Need circuits to manipulate 1’s and 0’s, they are called logic function A,B, C, Y – logic variables (binary signals) A, B,C – inputs Y – output Logic function: Y = F(A,B,C) A B C Y F
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ELEC151 Spring 2011 – L. Yobas Lecture 2 – 4 Truth Table Tabulate all possible input combinations of the variables Showing the relation between the values that the variables may take and the result of the operation E.g. ABCY 0001 0011 0100 0111 1000 1011 1101 1110 How can we design a circuit that implement a certain function specified by a truth table?
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ELEC151 Spring 2011 – L. Yobas Lecture 2 – 5 Boolean Algebra In 1854, a mathematician, George Boole, in a book called “An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic an Probabilities” developed an algebraic system to handle only two variables, TRUE and FALSE
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ELEC151 Spring 2011 – L. Yobas Lecture 2 – 6 Structure of Boolean Algebra a set of elements B Two binary operations {AND, OR} { , + } One unary operation {NOT} { ' } Axioms (Huntington Postulates) 1. Set B contains at least two elements, a , b , such that a b consider a two valued Boolean algebra, i.e. B = {0,1} 2. Closure (with respect to the Binary operator +, ) : (i) a + b is in B (ii) a b is in B 3. Commutative Laws : (i) a + b = b + a (ii) a b = b a 4. Associative Laws (i) a + ( b + c ) = ( a + b ) + c (ii) a • ( b c ) = ( a b) c 5. Identity elements with respect to the Binary operator : 0, 1 in B (i) a + 0 = a (ii) a • 1 = a 6. Distributive Laws : (i) a + ( b c ) = ( a + b ) • ( a + c ) (ii) a • ( b + c ) = a b + a c 7. Complement : (i) a + a' = 1 (ii) a a' = 0
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ELEC151 Spring 2011 – L. Yobas Lecture 2 – 7 Laws of Boolean Algebra Duality: A dual of a Boolean expression is derived by interchanging OR and AND operations, and 0s and 1s (literals are left unchanged) Any law that is true for an expression is also true for its dual. Laws (Theorems) of Boolean Algebra Operations with 0 and 1: x + 0 = x x 1 = x x + 1 = 1 x 0 = 0 Idempotent Law: x + x = x x x = x Involution Law: (x’)’ = x Laws of Complementarity: x + x’ = 1 x x’ = 0 Commutative Law: x + y = y + x x y = y x Associative Laws: (x + y) + z = x + (y + z) (x y) z = x (y z)
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ELEC151 Spring 2011 – L. Yobas Lecture 2 – 8 Laws of Boolean Algebra (Cont.) Distributive Laws: x (y + z) = (x y) + (x z) x +(y z) = (x + y)(x + z)
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Lecture 2- Boolean - ELEC151 Spring 2011 L.Yobas Lecture 2...

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