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Combinational Logic Circuits Chapter 2 Mano and Kime

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Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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Digital Logic Gates *

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Gates with More than Two Inputs

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Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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Basic Identities of Boolean Algebra

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Implementation of Boolean Function with Gates

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Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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Minterms for Three Variables

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Sum of Products Design X Y minterms 0 0 m0 = !X & !Y 0 1 m1 = !X & Y 1 0 m2 = X & !Y 1 1 m3 = X & Y

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Sum of Products Design X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 Design an XOR gate m1 = !X & Y m2 = X & !Y Z = m1 + m2 = (!X & Y) + (X & !Y)

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Sum of Products: Exclusive-OR !X & Y X & !Y Z = (!X & Y) + (X & !Y)

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Maxterms for Three Variables

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Product of Sums Design Maxterms: A maxterm is NOT a minterm maxterm M0 = NOT minterm m0 M0 = m0’ =(X’. Y’)’ = (X + Y)” = X + Y

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Product of Sums Design X Y minterms maxterms 0 0 m0 = !X. !Y M0 = !m0 = X + Y 0 1 m1 = !X. Y M1 = !m1 = X + !Y 1 0 m2 = X. !Y M2 = !m2 = !X + Y 1 1 m3 = X. Y M3 = !m3 = !X + !Y

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Product of Sums Design X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 Design an XOR gate Z is NOT minterm m0 AND it is NOT minterm m3

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Product of Sums Design X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 Design an XOR gate M0 = X + Y M3 = !X + !Y Z = M0 & M3 = (X + Y) & (!X + !Y)

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Product of Sums: Exclusive-OR

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Three- Level and Two- Level Implementation

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Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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Two-Variable Map

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Three-Variable Map

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Three- Variable Map: Flat and on a Cylinder to Show Adjacent Squares

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Three-variable K-Maps X YZ 00011110 0 1 11 11 F = !X & !Y + X & Z

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Three-variable K-Maps X YZ 00011110 0 1 11 11 F = !X & !Y & !Z + !X & !Y & Z + X & !Y & Z + X & Y & Z F = !X & !Y & (!Z + Z) + X & Z & (!Y + Y) = !X & !Y + X & Z

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Three-variable K-Maps X YZ 00011110 0 1 1 1 11 F = Y & !Z + X 1

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Three-variable K-Maps X YZ 00011110 0 1 11 111 1 F = !X & !Y + X & y + Z

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Three-variable K-Maps X YZ 00011110 0 1 11 11 F = X & Z + !X & !Z

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Three-variable K-Maps X YZ 00011110 0 1 11 11 1 1 F = Y + !Z

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Three-variable K-Maps X YZ 00011110 0 1 0123 4567 11 11 F = m0 + m2 + m5 + m7 = (0,2,5,7)

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Four-Variable Map

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Four-Variable Map: Flat and on a Torus to Show Adjacencies

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Four-variable K-Maps WX YZ 00011110 00 01 11 10 0 1 3 2 4 5 7 6 8 9 13 15 14 11 10 12 Each square is numbered in the above K-map

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Four-variable K-Maps WX YZ 00011110 00 01 11 10 0123 4567 89 11 12131415 F(W,X,Y,Z) = (2,4,5,6,7,9,13,14,15)

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Four-variable K-Maps 111 1 1 WX YZ 00011110 00 01 11 10 111 1 F = !W & X + X & Y + !W & Y & !Z + W & !Y & Z

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Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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Prime Implicants F = XY’Z + X’Z’ + X’Y Each product term is an implicant A product term that cannot have any of its variables removed and still imply the logic function is called a prime implicant.

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Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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Digital Logic Gates >

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>

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Logical Operations with NAND Gates

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Alternative Graphics Symbols for NAND and NOT Gates

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Logical Operations with NOR Gates

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Two Graphic Symbols for NOR Gate

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Generalized De Morgan’s Theorem NOT all variables Change & to + and + to & NOT the result -------------------------------------------- F = X & Y + X & Z + Y & Z F = !((!X + !Y) & (!X + !Z) & (!Y + !Z)) F = !(!(X & Y) & !(X & Z) & !(Y & Z))

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NAND Gate

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X Y X Z Y Z F F = X & Y + X & Z + Y & Z

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Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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Exclusive-OR Gate XOR X Y Z Z = X $ Y X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 X $ 0 = X X $ 1 = !X X $ X = 0 X $ !X = 1 X $ !Y = !(X $ Y) !X $ Y = !(X $ Y) A $ B = B $ A (A $ B) $ C = A $ (B $ C) = A $ B $ C

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Exclusive-OR Constructed with NAND gates X & (!X + !Y) + Y & (!X + !Y) = X & !X + X & !Y + Y & !X + Y & !Y = X & !Y + Y & !X = X & !Y + !X & Y = X $ Y

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Parity Generation and Checking

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Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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Fully Complementary CMOS Gate Structure and Examples An Integrated circuit (IC) is a silicon semiconductor crystal, containing the components for the digital gates. The various gates are connected on the chip to form the IC.

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ACOE1611 Combinational Logic Circuits Reference: M. Mano, C. Kime, “Logic and Computer Design Fundamentals”, Chapter 2.

ACOE1611 Combinational Logic Circuits Reference: M. Mano, C. Kime, “Logic and Computer Design Fundamentals”, Chapter 2.

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