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Unformatted text preview: 1/31/2012 1 Digital Logic Digital Logic and and Boolean Math Boolean Math CNIT 17600 Lecture 3 Digital Logic Digital logic refers to the process of determining an output based on a set of inputs Digital logic is often referred to as Boolean logic Having only two states True/On/High or False/Off/Low Digital logic is used at all levels of computing 1/31/2012 2 Boolean Algebra Boolean algebra is a mathematical system for the manipulation of variables that can have one of two values In formal logic, these values are true and false In digital systems, these values are on and off, 1 and 0, or high and low Boolean expressions are created by performing operations on Boolean variables AND OR NOT etc. 3 Logical AND The output is true only if both inputs are true Known as a Boolean product X and Y is denoted as XY AND Truth Table X Y XY 1 1 1 1 1 1/31/2012 3 Logical OR The output is true if either input is true Known as a Boolean sum X OR Y is denoted as X+Y OR Truth Table X Y X+Y 1 1 1 1 1 1 1 Logical NOT The output is the opposite of the input NOT X is denoted as X NOT Truth Table X X 1 1 1/31/2012 4 Boolean Functions A Boolean function has: At least one Boolean variable At least one Boolean operator At least one input from the set {0,1} It produces an output that is also a member of the set {0,1}. 7 The truth table for the Boolean function: 8 Boolean Algebra 1/31/2012 5 As with common arithmetic, Boolean operations have rules of precedence NOT AND OR This is how we chose the (shaded) function subparts in our table. 9 Boolean Algebra Simplifying Boolean Expressions Digital computers contain circuits that implement Boolean functions The simpler the Boolean function, the smaller the circuit Simpler circuits are cheaper to build, consume less power, and run faster than complex circuits The goal is to reduce Boolean functions to the simplest form Boolean identities help do this 10 1/31/2012 6 Basic Boolean Identities 11 Boolean Identities Basic Algebraic laws work here too Commutative Associative Distributive 12 1/31/2012 7 Boolean Identities Set theory rules apply as well 13 Boolean Identities Boolean identities can greatly simplify formulas: 14 1/31/2012...
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 Spring '09
 Hansen

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