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Princess Sumaya Univ. Computer Engineering Dept. Chapter 2:

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 1 / 28 Basic Definitions B inary Operators AND z = x y = x yz=1 if x=1 AND y=1 OR z = x + yz=1 if x=1 OR y=1 NOT z = x = x z=1 if x=0 B oolean Algebra Binary Variables: only 0 and 1 values Algebraic Manipulation

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 2 / 28 Boolean Algebra Postulates C ommutative Law x y = y xx + y = y + x I dentity Element x 1 = x x + 0 = x C omplement x x = 0 x + x = 1

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 3 / 28 Boolean Algebra Theorems D uality The dual of a Boolean algebraic expression is obtained by interchanging the AND and the OR operators and replacing the 1s by 0s and the 0s by 1s. x ( y + z ) = ( x y ) + ( x z ) x + ( y z ) = ( x + y ) ( x + z ) T heorem 1 x x = xx + x = x T heorem 2 x 0 = 0x + 1 = 1 Applied to a valid equation produces a valid equation

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 4 / 28 Boolean Algebra Theorems Theorem 3: Involution ( x ) = x ( x ) = x Theorem 4: Associative & Distributive ( x y ) z = x ( y z )( x + y ) + z = x + ( y + z ) x ( y + z ) = ( x y ) + ( x z ) x + ( y z ) = ( x + y ) ( x + z ) Theorem 5: DeMorgan ( x y ) = x + y ( x + y ) = x y Theorem 6: Absorption x ( x + y ) = x x + ( x y ) = x

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 5 / 28 Operator Precedence Parentheses (... ) (...) NOT x + y AND x + x y OR

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 6 / 28 DeMorgans Theorem

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 7 / 28 Boolean Functions B oolean Expression Example:F = x + y z T ruth Table All possible combinations of input variables L ogic Circuit xyzF 0000 0011 0100 0110 1001 1011 1101 1111

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 8 / 28 Algebraic Manipulation L iteral: A single variable within a term that may be complemented or not. U se Boolean Algebra to simplify Boolean functions to produce simpler circuits Example: Simplify to a minimum number of literals F = x + x y( 3 Literals) = x + ( x y ) = ( x + x ) ( x + y ) = ( 1 ) ( x + y ) = x + y( 2 Literals) Distributive law (+ over )

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 9 / 28 Complement of a Function DeMorgans Theorm Duality & Literal Complement

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 10 / 28 Canonical Forms M interm Product (AND function) Contains all variables Evaluates to 1 for a specific combination Example A = 0 A B C B = 0(0) (0) (0) C = 0 1 1 1 = 1 A B CMinterm 00 0 0m0m0 10 0 1m1m1 20 1 0m2m2 30 1 1m3m3 41 0 0m4m4 51 0 1m5m5 61 1 0m6m6 71 1 1m7m7

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 11 / 28 Canonical Forms M axterm Sum (OR function) Contains all variables Evaluates to 0 for a specific combination Example A = 1 A B C B = 1(1) + (1) + (1) C = 1 0 + 0 + 0 = 0 A B CMaxterm 00 0 0M0M0 10 0 1M1M1 20 1 0M2M2 30 1 1M3M3 41 0 0M4M4 51 0 1M5M5 61 1 0M6M6 71 1 1M7M7

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 12 / 28 Canonical Forms Truth Table to Boolean Function A B CF 0 0 00 0 0 11 0 1 00 0 1 10 1 0 01 1 0 11 1 1 00 1 1 11

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 13 / 28 Canonical Forms Sum of Minterms Product of Maxterms A B CF 00 0 00 10 0 11 20 1 00 30 1 10 41 0 01 51 0 11 61 1 00 71 1 11 F 1 0 1 1 0 0 1 0

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 14 / 28 Standard Forms Sum of Products (SOP)

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 15 / 28 Standard Forms Product of Sums (POS)

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 16 / 28 Two - Level Implementations Sum of Products (SOP) Product of Sums (POS)

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 17 / 28 Logic Operators AND NAND (Not AND) xyNAND 001 011 101 110 xyAND 000 010 100 111

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 18 / 28 Logic Operators OR NOR (Not OR) xyOR 000 011 101 111 xyNOR 001 010 100 110

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 19 / 28 Logic Operators XOR (Exclusive-OR) XNOR (Exclusive-NOR) (Equivalence) xyXOR 000 011 101 110 xyXNOR 001 010 100 111

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 20 / 28 Logic Operators NOT (Inverter) Buffer xNOT 01 10 xBuffer 00 11

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 21 / 28 Multiple Input Gates

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 22 / 28 DeMorgans Theorem on Gates AND Gate F = x yF = (x y) F = x + y OR Gate F = x + yF = (x + y) F = x y Change the Shape and bubble all lines

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 23 / 28 Homework Mano Chapter 2 2-4 2-5 2-6 2-8 2-9 2-10 2-12 2-15 2-18 2-19

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 24 / 28 Homework Mano 2-4Reduce the following Boolean expressions to the indicated number of literals: (a) AC + ABC + ACto three literals (b) (xy + z) + z + xy + wzto three literals (c) AB (D + CD) + B (A + ACD)to one literal (d) (A + C) (A + C) (A + B + CD)to four literals 2-5Find the complement of F = x + yz; then show that FF = 0 and F + F = 1

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 25 / 28 Homework 2-6Find the complement of the following expressions: (a) xy + xy(b) (AB + C)D + E (c) (x + y + z) (x + z) (x + y) 2-8List the truth table of the function: F = xy + xy + yz 2-9Logical operations can be performed on strings of bits by considering each pair of corresponding bits separately (this is called bitwise operation). Given two 8-bit strings A = 10101101 and B = 10001110, evaluate the 8-bit result after the following logical operations: (a) AND, (b) OR, (c) XOR, (d) NOT A, (e) NOT B.

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 26 / 28 Homework 2-10Draw the logic diagrams for the following Boolean expressions: (a) Y = AB + B (A + C)(b) Y = BC + AC (c) Y = A + CD(d) Y = (A + B) (C + D) 2-12Simplify the Boolean function T 1 and T 2 to a minimum number of literals. ABCT1T1 T2T2 00010 00110 01010 01101 10001 10101 11001 11101

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 27 / 28 Homework 2-15Given the Boolean function F = xyz + xyz + wxy + wxy + wxy (a) Obtain the truth table of the function. (b) Draw the logic diagram using the original Boolean expression. (c) Simplify the function to a minimum number of literals using Boolean algebra. (d) Obtain the truth table of the function from the simplified expression and show that it is the same as the one in part (a) (e) Draw the logic diagram from the simplified expression and compare the total number of gates with the diagram of part (b).

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Princess Sumaya University 4241 – Digital Logic Design Computer Engineering Dept. 28 / 28 Homework 2-18Convert the following to the other canonical form: (a) F (x, y, z) = (1, 3, 7) (b) F (A, B, C, D) = (0, 1, 2, 3, 4, 6, 12) 2-19Convert the following expressions into sum of products and product of sums: (a) (AB + C) (B + CD) (b) x + x (x + y) (y + z)

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