Boolean Algebra - Etsu

More documents

Recommendations

Info

$Practice for Final Exam - MATH 1720 Precalculus -- Fall 2003 - Etsu$

92 Computer Organization and Design Fundamentalstwo-input XOR equals one if A = 0 and B = 1 or if A = 1 and B = 0.This gives us the following expression.X = A ⊕ B = A·B + A·BThe next section shows how the boolean operators ·, +, ⊕, and theNOT bar may be combined to represent complex combinational logic.5.3 Boolean Expressions of Combinational LogicJust as mathematical algebra combines multiplication and additionto create complex expressions, boolean algebra combines AND, OR,and NOT functions to represent complex combinational logic. Ourexperience with algebra allows us to understand the expressionY = X · (X +5) + 3. The decimal value 5 is added to a copy of X, theresult of which is then multiplied by a second copy of X. Lastly, adecimal 3 is added and the final result is assigned to Y.This example shows us two things. First, each mathematicaloperation has a priority, e.g., multiplication is performed beforeaddition. This priority is referred to as precedence. Second, variablessuch X can appear multiple times in an expression, each appearancerepresenting the current value of X.Boolean algebra allows for the same operation. Take for examplethe circuit shown in Figure 5-6.ABCXFigure 5-6 Sample of Multi-Level Combinational LogicIn Chapter 4, we determined the truth table for this circuit by takingthe input signals A, B, and C from left to right through each gate. Asshown in Figure 5-7, we can do the same thing to determine theboolean expression.Notice the use of parenthesis in step c. Just as in mathematicalalgebra, parenthesis can be used to force the order in which operationsare taken. In the absence of parenthesis, however, the AND, OR, andNOT functions have an order of precedence.
Chapter 5: Boolean Algebra 93ABCABCBBA · BXXa) B goes through thefirst inverter whichoutputs a Bb) A and B go throughthe AND gatewhich outputsA · B.ABCBA · B(A · B) + CXc) A · B and C gothrough the ORgate which outputs(A · B) + C.ABCBA · B(A · B) + CX(A · B) + Cd) The output of theOR gate goesthrough a secondinverter giving usour result.Figure 5-7 Creating Boolean Expression from Combinational LogicTo begin with, AND takes precedence over OR unless overridden byparenthesis. NOT is a special case in that it can act like a set ofparenthesis. If the bar indicating the NOT function spans a singlevariable, it takes precedence over AND and OR. If, however, the NOTbar spans an expression, the expression beneath the bar must beevaluated before the NOT is taken. Figure 5-8 presents two examplesof handling precedence with the NOT function.ABX = A · BABX = A · BFigure 5-8 Examples of the Precedence of the NOT Function
Page 1 and 2: CHAPTER FIVEBoolean Algebra5.1 Need
Page 3: Chapter 5: Boolean Algebra 91functi
Page 7 and 8: Chapter 5: Boolean Algebra 95The fo
Page 9 and 10: Chapter 5: Boolean Algebra 97expres
Page 11 and 12: Chapter 5: Boolean Algebra 99gate o
Page 13 and 14: Chapter 5: Boolean Algebra 101Solut
Page 15 and 16: Solution_ _ _ _A·A + A·B + B·A +
Page 17 and 18: Chapter 5: Boolean Algebra 105switc
Page 19 and 20: Chapter 5: Boolean Algebra 107optim

Boolean Algebra - Etsu

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?