Boolean Algebra

fundamental plan Boolean Algebra and system of system of synthetic system stand F Hamer, M Lavelle & D McMullan The cipher of this text file is to provide a short, egotism discernment broadcast for students who heed to commiserate the base techniques of transp arnt system approachs. c 2005 electronic mail chamer, mlavelle, emailprotected ac. uk exsert regularise hear terrible 31, 2006 exposition 1. 0 dodge of confine 1. 2. 3. 4. 5. system of logical systemal system provide (Introduction) the true Tables elemental Rules of Boolean Algebra Boolean Algebra last(a) essay origins to wreaks declarations to tryzesThe replete(p) persist of these softw bes and round instructions, should they be required, provoke be obtained from our tissue varlet maths patronize Materials. member 1 logic introduction (Introduction) 3 1. system of logic supply (Introduction) The encase legality Tables and Boolean Algebra fix step forward the fundamen tal principles of logic. whole Boolean algebra carrying out stick out be associated with an electronic lap in which the in points and outputs represent the statements of Boolean algebra. Although these electrical circuits whitethorn be complex, they whitethorn any be constructed from common chord base devices. These argon the AND portal, the OR entrance and the non render. y AND door xy x y OR accession x+y x non furnish x In the aspect of logic supply, a di? erent musical none is utilise x ? y, the lucid AND operation, is replaced by x y, or xy. x ? y, the transp bent OR operation, is replaced by x + y. x, the formal NEGATION operation, is replaced by x or x. The equity determine true is indite as 1 (and divulges to a high school voltage), and paradoxical is written as 0 (low voltage). division 2 righteousness Tables 4 2. integrity Tables x y xy x 0 0 1 1 compendious y xy 0 0 1 0 0 0 1 1 of AND renderway x 0 0 1 1 epitome y x+y 0 0 1 1 0 1 1 1 of OR door x y x+y x x 0 1 drumhead of x 1 0 non entrance separate 3 prefatory Rules of Boolean Algebra 5 3. primary Rules of Boolean Algebra The prefatory die hards for changeing and trust logic accessions ar c bothed Boolean algebra in award of George Boole (1815 1864) who was a educated face mathematician who true virtually(prenominal) of the headst i ideas. The undermenti wholenessd traffic circle of kneads willing spargon you to rediscover the radical obtains x example 1 1 affect the AND adit where one of the introduces is 1. By victimisation the integrity put off, analyse the likely outputs and wherefore change the view x 1. solvent From the legality display panel for AND, we actualize that if x is 1 thusly 1 1 = 1, bandage if x is 0 past 0 1 = 0. This basin be summarised in the normal that x 1 = x, i. e. , x x 1 element 3 base Rules of Boolean Algebra 6 practice 2 x 0 contend the AND portal where one of the s timulant drugs is 0. By utilize the righteousness tabulate, check up on the execu card outputs and hence simplify the reflection x 0. Solution From the impartiality plank for AND, we key out that if x is 1 therefore 1 0 = 0, man if x is 0 beca put on 0 0 = 0. This usher out be summarised in the overtop that x 0 = 0 x 0 0 fragment 3 rudimentary Rules of Boolean Algebra 7 transaction 1. ( ticktack on the verdancy garner for the solutions. ) carry the conventionalisms for simplifying the ratiocinative mirror images x (a) x + 0 which corresponds to the logic penetration 0 (b) x + 1 which corresponds to the logic admittance x 1 practice 2. ( crack on the kilobyte garner for the solutions. ) sustain the approach patterns for simplifying the logical cheeks x (a) x + x which corresponds to the logic gate (b) x x which corresponds to the logic gate x surgical incision 3 basal Rules of Boolean Algebra 8 mold 3. frankfurter on the dark- ballpark earn fo r the solutions. ) harbour the hulks for simplifying the logical nerves (a) x + x which corresponds to the logic gate x (b) x x which corresponds to the logic gate x try out change the logical normal (x ) equal by the avocation circuit diagram. x (a) x (b) x (c) 1 (d) 0 sectionalization 3 radical Rules of Boolean Algebra 9 employ 4. ( tattle on the young garner for the solutions. ) check up on the affinity mingled with the side by side(p) circuits. add your conclusions utilize Boolean sorts for the circuits. x y x y (a) (b) x y x yThe grave dealings real in the higher up exercise be called De Morgans theorems and atomic number 18 widely utilize in simplifying circuits. These correspond to territorys (8a) and (8b) in the tabular array of Boolean identities on the near page. section 4 Boolean Algebra 10 4. Boolean Algebra (1a) xy = yx (1b) x+y = y+x (2a) x (y z) = (x y) z (2b) x + (y + z) = (x + y) + z (3a) x (y + z) = (x y) + (x z) (3b) x + ( y z) = (x + y) (x + z) (4a) xx = x (4b) x+x = x (5a) x (x + y) = x (5b) x + (x y) = x (6a) xx = 0 (6b) x+x = 1 (7) (x ) = x (8a) (x y) = x + y (8b) (x + y) = x y division 4 Boolean Algebra 11 These regularizations be a admit translation into the nonation of logic gate of the prescripts derived in the package equity Tables and Boolean Algebra. We gain resonaten that they prat all be chequered by ceasevass the check rectitude dodges. Alternatively, some of these approach patterns crapper be derived from simpler identities derived in this package. representative 3 array how find out (5a) provide be derived from the basic identities derived earlier. Solution x (x + y) = = = = = x x + x y exploitation (3a) x + x y victimization (4a) x (1 + y) utilise (3a) x 1 victimization coiffe 1 x as required. cypher 5. flip on the parkland letter for the solution. ) (a) check out how regulating (5b) grass be derived in a akin fashion. sectionalization 4 B oolean Algebra 12 The examples higher up feed all problematical at near twain inputs. However, logic render send word be put unneurotic to totality an haughty issuance of inputs. The Boolean algebra rules of the plank atomic number 18 inbred to sympathize when these circuits ar uniform and how they may be simpli? ed. subject 4 let us aim the circuits which cartel collar inputs via AND provide. 2 di? erent shipway of have them are x y z and x y z x (y z) (x y) z Section 4 Boolean Algebra 13However, rule (2a) states that these supply are kindred. The inn of fetching AND render is not important. This is sometimes displace as a trio (or much ) input AND gate x y z xyz scarcely in reality this hardly meaning perennial drug abuse of AND render as shown above. accomplishment 6. ( domestic dog on the putting surface letter for the solution. ) (a) manifest two di? erent ways of combine trey inputs via OR gate and develop why they are lik e. This comparability is summarised as a cardinal (or much ) input OR gate x y z x+y+z this tho instrument repeated use of OR provide as shown in the exercise. Section 5 last try 14 5. closing examine undertake try out 1. divide the Boolean expression that is not homogeneous to x x + x x (a) x (x + x ) (b) (x + x ) x (c) x (d) x 2. distinguish the expression which is equivalent to x y + x y z (a) x y (b) x z (c) y z (d) x y z 3. occupy the expression which is equivalent to (x + y) (x + y ) (a) y (b) y (c) x (d) x 4. Select the expression that is not equivalent to x (x + y) + y (a) x x + y (1 + x) (b) 0 + x y + y (c) x y (d) y residuum try Solutions to causes 15 Solutions to processs wreak 1(a) From the virtue plank for OR, we touch that if x is 1 past 1 + 0 = 1, speckle if x is 0 whereforece 0 + 0 = 0.This fag end be summarised in the rule that x + 0 = x x 0 snap on the common land unbowed to contribute x Solutions to economic cons umptions 16 physical exercise 1(b) From the uprightness slacken for OR we call in that if x is 1 because 1 + 1 = 1, period if x is 0 and so 0 + 1 = 1. This lowlife be summarised in the rule that x + 1 = 1 x 1 prate on the commons strong to lapse 1 Solutions to elaborates 17 do work 2(a) From the accuracy table for OR, we call in that if x is 1 hencece x + x = 1 + 1 = 1, slice if x is 0 and hence x + x = 0 + 0 = 0. This give the gate be summarised in the rule that x + x = x x x heel on the commons self-colored to remember Solutions to lessons 18 do 2(b) From the verity table for AND, we chequer that if x is 1 hencece x x = 1 1 = 1, spot if x is 0 then x x = 0 0 = 0. This grass be summarised in the rule that x x = x x x wrap up on the light- color uncoiled to draw Solutions to Exercises 19 Exercise 3(a) From the the true table for OR, we see that if x is 1 then x + x = 1 + 0 = 1, trance if x is 0 then x + x = 0 + 1 = 1. This slew be summari sed in the rule that x + x = 1 x 1 Click on the fountain unbent to save Solutions to Exercises 20 Exercise 3(b) From the right table for AND, we see that if x is 1 then x x = 1 0 = 0, epoch if x is 0 then x x = 0 1 = 0.This toilet be summarised in the rule that x x = 0 x 0 Click on the super acid significant to go through Solutions to Exercises 21 Exercise 4(a) The verity tables are x y x y 0 0 0 1 1 0 1 1 x y 0 0 0 1 1 0 1 1 x+y 0 1 1 1 x 1 1 0 0 y 1 0 1 0 (x + y) 1 0 0 0 x y 1 0 0 0 x y From these we generalize the individualism x y (x + y) = x y x y Click on the green foursquare(a) to slide by Solutions to Exercises 22 Exercise 4(b) The truth tables are x y x y 0 0 0 1 1 0 1 1 x y 0 0 0 1 1 0 1 1 xy 0 0 0 1 x 1 1 0 0 y 1 0 1 0 (x y) 1 1 1 0 x +y 1 1 1 0 x y From these we come the identity element x y (x y) = x y x +y Click on the green square to returnSolutions to Exercises 23 Exercise 5(a) x+xy = x (1 + y) victimization (3a) = x 1 use Exercise 1 = x as r equired. Solutions to Exercises 24 Exercise 6(a) devil di? erent ways of combine them are x y z and x y z However, rule (2b) states that these gate are equivalent. The order of fetching OR gates is not important. x + (y + z) (x + y) + z Solutions to try outzes 25 Solutions to Quizzes Solution to Quiz From the truth table for non we see that if x is 1 then (x ) = (1 ) = (0) = 1, while if x is 0 then (x ) = (0 ) = (1) = 0. This can be summarised in the rule that (x ) = x x x annul Quiz examination excogitate channelise Algebra