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# Boolean algebra

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### Boolean algebra

1. 1. Points Addressed in this Lecture • Theorems & rules in Boolean algebra • DeMorgan’s Theorems • Universality of NAND & NOR gates Lecture 4: Boolean Algebra • Active low & Active high Professor Peter Cheung • Digital Integrated Circuits Department of EEE, Imperial College London (Floyd 4.1-4.4, 5.2-5.4) (Tocci 3.8-3.14)E1.2 Digital Electronics I 4.1 Oct 2007 E1.2 Digital Electronics I Oct 2007 Laws of Boolean Algebra Commutative Laws of Boolean Algebra • Commutative Laws A+B=B+A • Associative Laws • Distributive Law A•B=B•AE1.2 Digital Electronics I 4.3 Oct 2007 E1.2 Digital Electronics I 4.4 Oct 2007
2. 2. Associative Laws of Boolean Algebra Distributive Laws of Boolean Algebra A • (B + C) = A • B + A • C A + (B + C) = (A + B) + C A (B + C) = A B + A C A • (B • C) = (A • B) • CE1.2 Digital Electronics I 4.5 Oct 2007 E1.2 Digital Electronics I 4.6 Oct 2007 Rules of Boolean Algebra Rules of Boolean Algebra • Rule 1 OR Truth Table • Rule 2 • Rule 3 AND Truth Table • Rule 4E1.2 Digital Electronics I 4.7 Oct 2007 E1.2 Digital Electronics I 4.8 Oct 2007
3. 3. Rules of Boolean Algebra Rules of Boolean Algebra• Rule 5 • Rule 9 • Rule 10: A + AB = A OR Truth Table• Rule 6 AND Truth Table OR Truth Table • Rule 7 AND Truth Table • Rule 8E1.2 Digital Electronics I 4.9 Oct 2007 E1.2 Digital Electronics I 4.10 Oct 2007 Rules of Boolean Algebra Examples • Rule 11: A + AB = A + B • Simplify the expression y = ABD + ABD y = AB • Simplify ( ) z = A + B (A + B ) z=B• Rule 12: (A + B)(A + C) = A + BC • Simplify x = ACD+ ABCD x = ACD + BCDE1.2 Digital Electronics I 4.11 Oct 2007 E1.2 Digital Electronics I 4.12 Oct 2007
4. 4. DeMorgan’s Theorems Review Questions • Theorem 1 y = AC + AB C• Simplify (x + y ) = x ⋅ y Remember: y = AC “Break the bar, • Theorem 2 change the operator”• Simplify y = ABC D + ABC D (x ⋅ y ) = x + y y = AB D – DeMorgans theorem is very useful in digital circuit design – It allows ANDs to be exchanged with ORs by using invertors• Simplify y = AD + ABD – DeMorgans Theorem can be extended to any number of variables. y = AD + BDE1.2 Digital Electronics I 4.13 Oct 2007 E1.2 Digital Electronics I 4.14 Oct 2007 Example of DeMorgan’s Theorem Example • Simplify the expression F = X .Y + P.Q 2 NAND plus 1 OR 1 OR with some input invertors ( )( z = A+ C ⋅ B + D ) = X +Y + P +Q to one having only single variables inverted. z = AC + B DE1.2 Digital Electronics I 4.15 Oct 2007 E1.2 Digital Electronics I 4.16 Oct 2007
5. 5. Implications of DeMorgan’s Theorems(I) Implications of DeMorgan’s Theorems(II)E1.2 Digital Electronics I 4.17 Oct 2007 E1.2 Digital Electronics I 4.18 Oct 2007 Example Review Questions • Determine the output expression for the below • Using DeMorgan’s Theorems to convert the circuit and simplify it using DeMorgan’s expressions to one that has only single-variable Theorem inversions. y = R ST + Q y = ( R + S + T )Q z = ( A + B )⋅ C z = AB + C • Use DeMorgan’s theorems to convert below expression to an expression containing only single-variable inversions. y = A + B + CD y = A B(C + D )E1.2 Digital Electronics I 4.19 Oct 2007 E1.2 Digital Electronics I 4.20 Oct 2007
6. 6. Universality of NAND gates Universality of NOR gate E1.2 Digital Electronics I 4.21 Oct 2007 E1.2 Digital Electronics I 4.22 Oct 2007 Basic Characteristics of Digital ICs• Digital ICs (chips): a collection of resistors, diodes and transistors fabricated on a single piece of semiconductor materials called substrate.• Dual-in-line package (DIP) is a common type of packages. Logic probe is used to monitor the logic level activity at an IC pin or any other accessible point in a logic circuit E1.2 Digital Electronics I 4.23 Oct 2007 E1.2 Digital Electronics I 4.24 Oct 2007
7. 7. ExampleE1.2 Digital Electronics I 4.25 Oct 2007 E1.2 Digital Electronics I 4.26 Oct 2007 Alternate Logic-Gate Representations Obtain alternative symbol from standard ones • Invert each input and output of the standard symbol. This is done by adding bubbles(small circles) on input and output lines that do not have bubbles and by removing bubbles that are already there. • Change the operation symbol from AND to OR, or from OR to AND. (In the special case of the INVERTER, the operation symbol is not changed.)E1.2 Digital Electronics I 4.27 Oct 2007 E1.2 Digital Electronics I 4.28 Oct 2007
8. 8. Several points Logic-symbol interpretation • The equivalences can be extended to gates with any number of inputs. • Active high/low • None of the standard symbols have bubbles on their – When an input or output line on a logic circuit inputs, and all the alternate symbols do. symbol has no bubble on it, that line is said to • The standard and alternate symbols for each gate be active-high, otherwise it is active-low. represent the same physical circuit; there is no difference in the circuits represented by the two symbols. • NAND and NOR gates are inverting gates, and so both the standard and the alternate symbols for each will have a bubble on either the input or the output, AND and OR gates are non-inverting gates, and so the alternate symbols for each will have bubbles on both inputs and output.E1.2 Digital Electronics I 4.29 Oct 2007 E1.2 Digital Electronics I 4.30 Oct 2007 Interpretation of the two NAND gate symbols Interpretation of the two OR gate symbolsE1.2 Digital Electronics I 4.31 Oct 2007 E1.2 Digital Electronics I 4.32 Oct 2007
9. 9. Equivalency of three circuits • Which circuit diagram should be used? • Bubble placement Whenever possible, choose gate symbols so that bubble outputs are connected to bubble inputs, and nonbubble outputs to nonbubble inputsE1.2 Digital Electronics I 4.33 Oct 2007 E1.2 Digital Electronics I 4.34 Oct 2007 Active High & Active Low Representations • When a logic signal is in its active state, it can be said to be asserted (=activated). When a logic signal is not in its active state, it is said to be unasserted (=inactive). • The overbar is simply a way to emphasize that these are active-LOW signals.E1.2 Digital Electronics I 4.35 Oct 2007 E1.2 Digital Electronics I 4.36 Oct 2007
10. 10. Summary: Summary How to represent the basic logic functions • Boolean Algebra: a mathematical tool used in the analysis and design of digital circuits• Logical statements in our own language • OR, AND, NOT: basic Boolean operations• Truth tables • OR: HIGH output when any input is HIGH • AND: HIGH output only when all inputs are HIGH• Traditional graphic logic symbols • NOT: output is the opposite logic level as the input• Boolean algebra expressions • NOR: OR with its output connected to an INVERTER• Timing diagrams • NAND: AND with its output connected to an INVERTER • Boolean theorems and rules: to simplify the expression of a logic circuit and can lead to a simpler way of implementing the circuit • NAND, NOR: can be used to implement any of the basic Boolean operationsE1.2 Digital Electronics I 4.37 Oct 2007 E1.2 Digital Electronics I 4.38 Oct 2007