02 Boolean Algebra

# 02 Boolean Algebra - EE2000 Logic Circuit Design 2 Boolean...

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1 EE2000 Logic Circuit Design 2 Boolean Algebra and Logic Gates ±±

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2 Outline ± 2.1 Concept of Boolean Algebra ± Truth table, Boolean function, circuit diagram ± 2.2 Basic Postulates of Boolean Algebra ± Basic properties, theorems, duality principle ± 2.3 Boolean Functions and Their Representation ± Minterm, maxterm, standard forms ± 2.4 Other Logic Gates ± NAND, NOR, XOR, XNOR
3 2.1 Concept of Boolean Algebra

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4 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 ± The design logic circuits would be simplified through the manipulation of Boolean expressions
5 Boolean Algebra ± Boolean algebra is an algebraic system consisting of ± The set of elements {0, 1} ± Two binary operators (OR, AND) ± One unary operator (NOT) ± Binary operator requires two operands ± Unary operator requires one operand only

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6 2.1.1 Logical Operators ± 3 basic logical operators OR AND NOT Binary / Unary operator? Binary Binary Unary Symbols 1: + 2: V 1: · 2: Λ 3: absence of an operator 1: 2: ~ 3: ¯ Examples 1: a + b 2: a V b 1: x · y 2: x Λ y 3: x y 1: a’ 2: ~ a 3: a Logic Gate Symbol a b x y f f a f
7 2.1.2 Binary Variables ± Binary variables (and constants) take on one of two truth values, 0 and 1 ± a = 0, 1 ± Binary variables arithmetic variables ± Arithmetic variables may consist of many digits ± a = -1, 0, 1, 2 …, 0.5, 1.2, …, 10 / 3, 22/7, …

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8 Logical Operators Definition ± OR ± 0 + 0 = 0 (read as 0 OR 0 , not 0 plus 0 ) ± 0 + 1 = 1 ± 1 + 0 = 1 ± 1 + 1 = 1 ± AND ± 0 · 0 = 0 (read as 0 AND 0 , not 0 times 0 ) ± 0 · 1 = 0 ± 1 · 0 = 0 ± 1 · 1 = 1 ± NOT ± 0’ = 1 (read as NOT 0 ) ± 1’ = 0
9 2.1.3 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 a b f 0 0 0 0 1 1 1 0 1 1 1 1 Input(s) Output All input combinations The corresponding output

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10 Truth Table of OR, AND, NOT ± There are other logical operations ± e.g. NOR, NAND, XOR, XNOR, etc ± Will re-visit them later a b a + b 0 0 0 0 1 1 1 0 1 1 1 1 Truth table of OR a b a · b 0 0 0 0 1 0 1 0 0 1 1 1 Truth table of AND a a 0 1 1 0 Truth table of NOT
11 2.1.3 Boolean Expression ± Boolean expression is an algebraic expression formed by binary variables , constants , logic operations and parentheses ± e.g. ± a + b’c’ ± a’b’ + bc’ + ( c + 1 · ab’c ) + d

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12 2.1.4 Boolean Function ± Boolean function is a Boolean equation consisting of binary variable followed by an equal sign and a Boolean expression ± e.g. x = a + b’c’ ± y = a’b’ + bc’ + ab’c + d ± Usually parentheses enclosed a list of function variable, separated by commas ± e.g. x ( a , b , c ) = a + b’ c’ ± e.g. y ( a , b , c , d ) = a’ b’ + b c’ + a b’ c + d
13 Relationship ± Relation between Boolean function, truth table and logic circuit diagram

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02 Boolean Algebra - EE2000 Logic Circuit Design 2 Boolean...

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