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Chapter 03 - Boolean Algebra II-2x2(1)

Chapter 03 - Boolean Algebra II-2x2(1) - Distributive Laws...

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CHAPTER # BOOLEAN ALGEBRA ± II )(0QQNNHHPP YMMJJ RRTTZXJJ YTT RRTT[JJ YTT YMMJJ SSJJ]Y UUFLLJJ° :XJJ YMMJJ +*28)(0 PPJJ^ YTT JJ]NNY YMMNNX HHMMFUUYJJW° °±² 2Multiplying Out and ++actoring **xpressions °±³ **xclusive´OR and **quivalence Operations °±° The (((onsensus Theorem °±µ &&&lgebraic Simplification of Switching **xpressions °±¶ Proving the Validity of an **quation +))/istributive 175aws ,,iven an expression in product´of´sums form· the corresponding sum´of´products expression can be obtained by multiplying out· using the two distributive laws$ X ¸ Y ± Z ¹ '""" XY º XZ ¸°´²¹ ¸ X ± Y ¹¸ X ± Z ¹ '""" X ± YZ ¸°´³¹ .n addition· the following theorem is very useful for factoring and multiplying out$ ¸ X ± Y ¹¸ X′ ± Z ¹ '""" XZ ± X ƍ Y ¸°´°¹ +*2]FRRUUQQJJ ±²³´µ¶ UU° !² .n the following example· if we were to multiply out by brute force· we would generate ² ³ terms· and ²¶" of these terms would then have to be eliminated to simplify the expression± .nstead· we will use the distributive laws to simplify the process± The same theorems that are useful for multiplying out expressions are useful for factoring± '''y repeatedly applying ¸°´²¹· ¸°´³¹· and ¸°´°¹· any expression can be converted to a product´of´sums form±
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,**0xclusive°:8=; and ,**0quivalence :8perations The exclusive°OR operation ¸ ¹ is defined as follows$ The equivalence operation ¸ ¹ is defined by$ Section "3±!2² p± %6#4 We will use the following symbol for an exclusive° :8=; gate$ The following theorems apply to exclusive OR$ Section "3±!2² p± %6$5 We will use the following symbol for an equivalence gate$
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Section "3±!2² p± %6%6 '''ecause equivalence is the complement of exclusive°:8=; · an alternate symbol of the equivalence gate is an exclusive°:8=; gate with a complemented output$ The equivalence gate is also called an exclusive°97:8=; gate± **xample ²$ **xample ³$ ><ection ±²³ ´p² !!µ '''y ¸°´ ¹ and ¸°´²!¹·
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