Boolean Algebra - Etsu

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

96 Computer Organization and Design FundamentalsA + (B + C) = (A + B) + CA · (B · C) = (A · B) · CDistributive Law: The AND'ing of an operand with an OR expressionis equivalent to OR'ing the results of an AND between the first operandand each operand within the OR expression.A · (B + C) = A · B + A · CThe next section uses truth tables and laws to prove twelve rules ofboolean algebra.5.5 Rules of Boolean Algebra5.5.1 NOT RuleIn algebra, the negative of a negative is a positive and taking theinverse of an inverse returns the original value. Although the NOT gatedoes not have an equivalent in mathematical algebra, it operates in asimilar manner. If the boolean inverse of a boolean inverse is taken, theoriginal value results.This is proven with a truth table.A = AA A A0 1 01 0 1Since the first column and the third column have the same pattern ofones and zeros, they must be equivalent. Figure 5-11 shows this rule inschematic form.AA = AFigure 5-11 Schematic Form of NOT Rule5.5.2 OR RulesIf an input to a logic gate is a constant 0 or 1 or if the same signal isconnected to more than one input of a gate, a simplification of the
Chapter 5: Boolean Algebra 97expression is almost always possible. This is true for the OR gate as isshown with the following four rules for simplifying the OR function.First, what happens when one of the inputs to an OR gate is aconstant logic 0? It turns out that the logic 0 input drops out leavingthe remaining inputs to stand on their own. Notice that the two columnsin the truth table below are equivalent thus proving this rule.Rule: A + 0 = AA A + 00 0+0 = 01 1+0 = 1What about inputting a logic 1 to an OR gate? In this case, a logic 1forces the other operands into the OR gate to drop out. Notice that theoutput column (A + 1) is always equal to 1 regardless of what A equals.Therefore, the output of this gate will always be 1.Rule: A + 1 = 1A A + 10 0+1 = 11 1+1 = 1If the same operand is connected to all of the inputs of an OR gate,we find that the OR gate has no effect. Notice that the two columns inthe truth table below are equivalent thus proving this rule.A A + ARule: A + A = A 0 0+0 = 01 1+1 = 1Another case of simplification occurs when an operand is connectedto one input of a two-input OR gate and its inverse is connected to theother. In this case, either the operand is equal to a one or its inverse is.There is no other possibility. Therefore, at least one logic 1 isconnected to the inputs of the OR gate. This gives us an output of logic1 regardless of the inputs.A A + ARule: A + A = 1 0 0+1 = 11 1+0 = 15.5.3 AND RulesJust as with the OR gate, if either of the inputs to an AND gate is aconstant (logic 0 or logic 1) or if the two inputs are the same or inverses
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Boolean Algebra - Etsu

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