02 Boolean Algebra - EE2000 Logic Circuit Design Boolean...

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1 EE2000 Logic Circuit Design Boolean Functions and Logic Gates
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2 Outline ± Boolean Algebra ± Logic Operations ± Properties ± Manipulating Algebraic Functions ± Min-term & Max-term Canonical Forms ± Sum of Products and Products of Sums ± DeMorgan’s Theorem ± Basic Logic Gates
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3 Boolean Algebra ± In 1854, George Boole introduced the mathematical theory of logic ± This binary logic system called Boolean Algebras ± Mathematical notation to describe the interconnection of digital gates ± To simplify the design logic circuits through the manipulation of Boolean expressions
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4 You are going to learn… ± Concept of binary logic ± Properties of Boolean algebra ± Manipulation of Boolean expression, and ± Logic gates
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5 Concept of Binary Logic
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6 Binary Logic ± Binary logic (Boolean algebra) deals with binary variables and mathematical logic operations ± Binary variables (and constants) take on one of two discrete values, 0 and 1 ± Examples ± False (0) or True (1) ± Light off (0) or on (1) ± Low voltage (0) or high voltage (1) ± Magnetic field in one direction or the other
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7 The Basic Logical Operators ± OR ± Binary Operator ± written as + , ν ± a + b is 1 if and only if a = 1 or b = 1, or both ± AND ± Binary Operator ± written as · , Λ , or the absence of an operator ± a · b = ab is 1 iff a = 1 and b = 1 ± NOT (inversion, complement) ± Unary Operator ± NOT a can written as a , ~ a , ā ± a is 1 iff a = 0
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8 Why Use · and + symbols? ± AND , OR operations have similarities to multiplication and addition respectively ± But don’t confuse binary logic with binary arithmetic ± Logic variables are always either 1 or 0 ± Arithmetic variables may consist of many digits
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9 Logical OR operation ± 0 + 0 = 0 (read as 0 OR 0 , not 0 plus 0 ) ± 0 + 1 = 1 ± 1 + 0 = 1 ± 1 + 1 = 1 ± Resemble binary addition (except for the last operation) ± In binary arithmetic, 1 + 1 = 10 (read as 1 plus 1 = 2 )
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10 Logical AND operation ± 0 · 0 = 0 (read as 0 AND 0 , not 0 dot 0 ) ± 0 · 1 = 0 ± 1 · 0 = 0 ± 1 · 1 = 1 ± Identical to binary multiplication ± The above defined all combinations of the values of a logical operation ± List the definition in a compact form – truth table
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11 Truth table ± A table of combinations of the binary variables showing the relationship between the values take on and the values of the result of the operation 1 1 1 1 0 1 1 1 0 0 0 0 a + b b a Truth table (OR) 1 1 1 0 0 1 0 1 0 0 0 0 a · b b a Truth table (AND) 0 1 1 0 a a Truth table (NOT)
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12 Boolean Algebra & Expression ± Boolean algebra is an algebra dealing with binary variables and logic operations ± e.g. b · c’ ± Boolean expression is an algebraic expression formed by binary variables , constants , logic operations and parentheses ± e.g. a’b’ + bc’ + (1 · ab’c )
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13 Boolean Function ± Boolean function is a Boolean equation consisting of binary variable followed by an equal sign and a Boolean expression ± e.g. z = a’ b’ + b c’ + a b’ c + d ± Express the logical relationship between binary variables ± Usually parentheses enclosed a list of function variable, separated by commas ± e.g. f ( a , b , c , d ) = a’ b’ + b c’ + a b’ c + d
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This note was uploaded on 01/11/2011 for the course EE 2000 taught by Professor Vancwting during the Fall '07 term at City University of Hong Kong.

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02 Boolean Algebra - EE2000 Logic Circuit Design Boolean...

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