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Boolean Algebra and Logic Gates COE 202 Digital Logic Design Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 2 Presentation Outline Boolean Algebra Boolean Functions and Truth Tables DeMorgan's Theorem Algebraic manipulation and expression simplification Logic gates and logic diagrams Minterms and Maxterms Sum-Of-Products and Product-Of-Sums
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 3 Boolean Algebra
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 4 Postulates of Boolean Algebra
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 5 AND, OR, and NOT Operators x yx·y 0 0 0 10 1 00 1 1 x yx+y 0 0 0 11 1 01 1 1 xx' 01 10
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 6 Boolean Functions
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 7 Truth Table x y z y' xy' x'x'zf = xy'+ x'z 0 0 010100 0 0 110111 0 1 000100 0 1 100111 1 0 011001 1 0 111001 1 1 000000 1 1 100000
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 8 DeMorgan's Theorem xyx'y'x+y(x+y)'x'y'x yx y(x y)'x'+ y' 0011011011 0110100011 1001100011 1100100100 Can be verified Using a Truth Table Identical
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 9 Complementing Boolean Functions
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 10 Algebraic Manipulation of Expressions
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 11 Consensus Theorem
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 12 Summary of Boolean Algebra PropertyDual Property Identity Complement Null Idempotence Involution Commutative Associative Distributive Absorption Simplification De Morgan
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 13 Duality Principle PropertyDual Property Identity Complement Distributive
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 14 Expression Simplification
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 15 Importance of Boolean Algebra Our objective is to learn how to design digital circuits These circuits use signals with two possible values Logic 0 is a low voltage signal (around 0 volts) Logic 1 is a high voltage signal (e.g. 5 or 3.3 volts) The physical value of a signal is the actual voltage it carries, while its logic value is either 0 (low) or 1 (high) Having only two logic values (0 and 1) simplifies the implementation of the digital circuit
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 16 Next... Boolean Algebra Boolean Functions and Truth Tables DeMorgan's Theorem Algebraic manipulation and expression simplification Logic gates and logic diagrams Minterms and Maxterms Sum-Of-Products and Product-Of-Sums
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 17 Logic Gates and Symbols AND gateOR gate NOT gate (inverter) In the earliest computers, relays were used as mechanical switches controlled by electricity (coils) Today, tiny transistors are used as electronic switches that implement the logic gates (CMOS technology) AND: Switches in series logic 0 is open switch OR: Switches in parallel logic 0 is open switch NOT: Switch is normally closed when x is 0
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 18 Truth Table and Logic Diagram Truth Table x y zy'+ zf = x(y'+ z) 0 0 010 0 0 110 0 1 000 0 1 110 1 0 011 1 0 111 1 1 000 1 1 111 Logic Diagram
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 19 Combinational Circuit Combinational Circuit
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 20 Example of a Simple Combinational Circuit
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 21 From Truth Tables to Gate Implementation Truth Table x y zf 0 0 00 0 0 10 0 1 01 0 1 11 1 0 00 1 0 11 1 1 00 1 1 11 To answer these questions, we need to define Minterms and Maxterms
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 22 Minterms and Maxterms xyindexMintermMaxterm 000 011 102 113
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 23 Minterms and Maxterms for 3 Variables xyzindex MintermMaxterm 0000 0011 0102 0113 1004 1015 1106 1117
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 24 Purpose of the Index Minterms and Maxterms are designated with an index The index for the Minterm or Maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true or complemented form For Minterms: ‘1’ means the variable is Not Complemented ‘0’ means the variable is Complemented For Maxterms: ‘0’ means the variable is Not Complemented ‘1’ means the variable is Complemented
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 25 Sum-Of-Minterms (SOM) Canonical Form Sum of Minterm entries that evaluate to ‘1’ Truth Table x y zfMinterm 0 0 00 0 0 10 0 1 01 0 1 11 1 0 00 1 0 11 1 1 00 1 1 11 Focus on the ‘1’ entries
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 26 Examples of Sum-Of-Minterms
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 27 Product-Of-Maxterms (POM) Canonical Form Truth Table x y zfMaxterm 0 0 00 0 0 10 0 1 01 0 1 11 1 0 00 1 0 11 1 1 00 1 1 11 Product of Maxterm entries that evaluate to ‘0’ Focus on the ‘0’ entries
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 28 Examples of Product-Of-Maxterms
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 29 Conversions between Canonical Forms x y zfMintermsMaxterms 0 0 01 0 0 10 0 1 01 0 1 11 1 0 00 1 0 11 1 1 00 1 1 11 To convert from one canonical form to another, interchange the symbols and and list those numbers missing from the original form. Truth Table
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 30 Function Complement x y zff' 0 0 010 0 0 101 0 1 010 0 1 110 1 0 001 1 0 110 1 1 001 1 1 110 The complement of a function expressed by a Sum of Minterms is the Product of Maxterms with the same indices. Interchange the symbols and , but keep the same list of indices. Truth Table
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 31 Summary of Minterms and Maxterms There are 2 n Minterms and Maxterms for Boolean functions with n variables, indexed from 0 to 2 n – 1 Minterms correspond to the 1-entries of the function Maxterms correspond to the 0-entries of the function Any Boolean function can be expressed as a Sum-of-Minterms and as a Product-of-Maxterms For a Boolean function, given the list of Minterm indices one can determine the list of Maxterms indices (and vice versa) The complement of a Sum-of-Minterms is a Product-of-Maxterms with the same indices (and vice versa)
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 32 Sum-of-Products and Products-of-Sums
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 33 Two-Level Gate Implementation 3-input AND gate AND-OR implementations 3-input OR gate OR-AND implementations
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Boolean Algebra and Logic GatesCOE 202 – Digital Logic Design© Muhamed Mudawar – slide 34 Two-Level vs. Three-Level Implementation 3-input OR gate
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