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Logic Design CS221 1 st Term 2009-2010 Boolean Algebra Cairo University Faculty of Computers and Information

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13/10/2009 cs221 – sherif khattab 2 Administrivia http://www.fci.cu.edu.eg/~skhattab/cs221 labs: download Atanua logic simulator from: http://atanua.org and bring it with you to the lab http://atanua.org download lab instructions (from course website) form groups of 3 in the lab project: use Atanua (display, audio, motors) design a digital circuit (e.g., simple video game, talking clock)

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13/10/2009 cs221 – sherif khattab 3 Bonus Chance Like a Bonus?? Find the error in the next slide (and in the book) before Saturday

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13/10/2009 cs221 – sherif khattab 4 boolean functions input -> output binary variables (and constants) and logic operations defined by a truth table for two variables, only 16 boolean functions possible. why? how many possible truth tables? for n variables, 2 2n boolean functions error

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13/10/2009 cs221 – sherif khattab 5 Boolean Algebra digital circuits implement Boolean functions Boolean algebra used by software tools to: simplify circuits into cheaper yet equivalent implementations (two-valued) Boolean algebra: elements (0, 1) operators (logic operators: AND, OR, NOT) defined by truth tables axioms (unproved ``assumptions”)

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13/10/2009 cs221 – sherif khattab 6 Boolean vs. Ordinary Algebra Differences: x + (y ∙ z) = (x + y) ∙ (x + z) inverse complement (NOT) Similarities: variables vs. elements

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13/10/2009 cs221 – sherif khattab 7 Basic Theorems Duality: if a statement is true using the AND operator => its dual is also true dual: interchange AND and OR and replace 0 with 1 and 1 with 0 Example: x + x = xdual: x ∙ x = x x + 0 = xdual: x ∙ 1 = x

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13/10/2009 cs221 – sherif khattab 8 Basic Axioms and Theorems

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13/10/2009 cs221 – sherif khattab 9 Basic Axioms and Theorems

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13/10/2009 cs221 – sherif khattab 10 Sample proof by truth table x y xyx + xy 0 0101 1010 1 0 0 0101 1

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13/10/2009 cs221 – sherif khattab 11 Boolean Functions F 1 = x + y'z algebraic form truth table logic-circuit diagram (schematic)

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13/10/2009 cs221 – sherif khattab 12 Boolean function simplification truth table: one way logic form and schematic: many ways look for simplest and cheapest reduce number of gates reduce number of inputs to gates Boolean expression simplification

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13/10/2009 cs221 – sherif khattab 13 Example F 2 = x'y'z + x'yz + xy'

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13/10/2009 cs221 – sherif khattab 14 Example (contd.) F 2 = x'y'z + x'yz + xy' = x'z(y' + y) + xy' = x'z ∙ 1 + xy' = x'z + xy'

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13/10/2009 cs221 – sherif khattab 15 reducing number of literals literal: a single variable within a term un-complemented or complemented F 2 = x'y'z + x'yz + xy' (8 literals) F 2 = x'z + xy' (?? literals)

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13/10/2009 cs221 – sherif khattab 16 simplification examples x(x' + y) = xx' + xy = xy x + x'y = (x + x')(x + y) = 1(x + y) = x + y (x + y)(x + y') = xx + xy' + xy + yy' = x + xy' + xy + 0 = x(1 + y' + y) = x

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13/10/2009 cs221 – sherif khattab 17 simplification examples xy + x'z + yz = xy + x'z + yz(x + x') = xy + x'z + xyz + x'yz = xy(1 + z) + x'z(1 + y) = xy + x'z (x + y)(x' + z)(y + z) = (x+y)(x' + z) (why?) consensus theorem

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13/10/2009 cs221 – sherif khattab 18 function complement interchange of 0's and 1's in truth table To get function complement: get dual then complement each literal (DeMorgan's theorem) F1'F1'

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13/10/2009 cs221 – sherif khattab 19 function complement examples F 1 = x'yz' + x'y'z dual: (x' + y + z')(x' + y' + z) complement literals: (x + y' + z)(x + y + z') = F 1 ' F 2 = x(y'z' + yz) dual: x+(y' + z')(y + z) complement of literals: x' + (y + z)(y' + z') = F 2 '

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13/10/2009 cs221 – sherif khattab 20 canonical and standard forms canonical forms sum of minterms product of maxterms standard forms sum of products product of sums

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13/10/2009 cs221 – sherif khattab 21 sum of minterms (1/2) minterms (standard products) minterms of 2 variables x and y x'y', x'y, xy', xy 2 n minterms of n variables

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13/10/2009 cs221 – sherif khattab 22 sum of minterms (2/2) from truth table f1 = m1 + m4 + m7 = ∑(1, 4, 7) = x'y'z + xy'z' + xyz f 2 = ??

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13/10/2009 cs221 – sherif khattab 23 product of maxterms (1/2) maxterms (standard sums) maxterms of 2 variables x and y x' + y', x' + y, x + y', x + y 2 n maxterms of n variables

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13/10/2009 cs221 – sherif khattab 24 product of maxterms (2/2) f1 = M 0 M 2 M 3 M 5 M 6 = ∏(0, 2, 3, 5, 6) = (x + y + z)(x + y' + z)(x + y' + z')(x' + y + z')(x' + y' + z) f 2 = ??

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13/10/2009 cs221 – sherif khattab 25 Conversion F(A, B, C) = ∑(1, 4, 5, 6, 7) 5 terms what is F in algebraic form? F(A, B, C) = ∏(0, 2, 3) 2 3 – 5 = 3 terms what happened? replaced ∑ by ∏ list missing terms

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13/10/2009 cs221 – sherif khattab 26 Standard forms (1/2) are canonical forms the simplest? sum of products: ex: F 1 = y' + xy + x'yz' two-level implementation

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13/10/2009 cs221 – sherif khattab 27 Standard forms (2/2) product of sums ex: F 2 = x(y' + z)(x' + y + z') two-level implementation

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13/10/2009 cs221 – sherif khattab 28 non-standard forms ex: F 3 = AB + C(D + E) three levels F 3 = AB + C(D + E) = AB + CD + CE which standard form? two levels which one is better?

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