Unit_4_SG - Study Guide in the previous units, we placed a...

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Unformatted text preview: Study Guide in the previous units, we placed a “-" inside of the AND—gate symbol. at "+" inside of the OR—gate symbol, and a “If-B" inside the Exclusive-OR. BecauSe you are now familiar with the relationship between the shape of the gate symbol and the logic function performed. we wit! omit the -. +. and 83 and use the standard gate symbols for AND. OR, and Exclusive— OR in me rest of the book. Study Section 4.1. C(Jrrverst'on rg‘i's'ugiirit Sentences w Boolean Equations: (it) Use braces to identify the phrases in flush of the following sentences: (I) The tape reader should stop if‘the manual stop hutton is Eressed. . 3 . . . .H . if an error occurs or It an cricket—tape signal is present. I--—-»————4 L——~»~——.._____J E .l. (2] He eats e tgs for breakfast ifit is not Sunday and I.._—.——..+.—_..__t E 5: Ihe has eggs in the rcfrigeratorr (3} Additiorfihould occur 'fl'an add instruction is Eivegzlnd 3’: 1 the signs are the saw: or if a subtract instruction is given and L—j—‘i‘ L—--—.____.~ K the signs are not the some, J' (it) Write a Beulah expression which represents each of the senlenees in (at). Assign a variable to each phrase. and use a compleinenled variable to represent a phrase which contains “not”. l- 5: M +T+E 2- E r: s'fl. 3. A -:. 15 + Xs’ (Youranswersshouidhein the FerrnF = S‘E. F = A3 + SB'.antl F = A + H + C. but not necessarily in that order.) to) Ierepi-esents the phrase “N is greater Thun 3". how can you represent the phrase “N is less man or equal to 3"? Kl to) Work Problems 4.] and 4.2. Study Section 4.2. Cont-Humane! Logic Design Using a Truth Tobie. Previously. you have learned how to go from an algebraic expression for tt function In :1 India table; in this section you will learn how to go from a truth table to an algebraic expression. t (:1) Write aproduetierrn which is 1 it'i'rr = I), b = CI. and e = 1. db C {h} Writeasum term which is O iffn = 0. b = G, andc =1. 0 +ia+cl to) Verify that your answers to (a) and (h) are complements. (stifle); n+b+c" (d) (a) {f} Applications of Boolean Algebra I Minterrn and Mat-(term Expansions 79 Write nprnduet term which is l iffn = 1. b = 0.6 = D. and 6‘ =1. nb'c' :1 Written sum term which isO iffa = 0,13 = {1e =.i. amid =1. {at-:- in: + c1+d') For the given truth table, write Fast a sum of four product terms which correspond to the four 1‘s of E o 0 g 1 F: a’b'e’+a'b'c+albc+ QL'e‘ OCH 1 01 E] (1' (g) me the truth table write Pas a product of four sum {1 1 1 1 terms which correspond to the fourfl's of E r 1 o 0. 1 F:(a+£=’+c) {u'+b+c')(ahb’+c3[a‘+B+CD 101 0 i 1 0 U (h) Verify that your answers to both (f) and Lg) reduce to 1 t ‘I U . . r r t F=bc +06. 1:: {a-rb-IC') fa‘+t2'+c)fg"-+ia+-c'i i t? = a'izg't 3. (a) {b} e'b’ei-e‘bswiis‘ : ' r | .1 h i t I EC (a'WIJ-E-q nib-Hail) -bct-GC Stud}.| Section 4.31 Minterm and Marta m: Expansions. ;Ltp'+-O[ct’+ct) : gaunt: Define the foilowing terms: minterrnUbrrtvuriablesj FrOdJud' 9‘9 In m Unuubie Duff'emin One: in 4+1: +114; o’r wwlemwi'ed 49min again ii maxterm (for n variabies) . Sum at in items. with w module. opgamu we»; at theme. nr cowth "Fawn in ant. teem Stud}! Table 4A] and observe the reiation between the values of :1, B, and C and the corresponding Ininterms and mnxterms. If)”. = D, then does A orfi' appear in Ihe mintern'l? A: in the maxterm? A MA = i, then doesA orA' appear in the minterrn'i l't 1n the maxterrn‘.’ A’ What is the rotation between minterm. m1. and the corresponding n‘mtttern‘t1 M? 3 . ML = “I For the table given in Study Guide Question 2(f), write the minterm expansion for F in tut-notation and in decimal notation. F: martnlrrtgfin.’I 2Mfifi,l,3,0r) For the same table, write the maxterrn expansion for F in iii-notation and in decimal notation. ravine Mei-t1 : Ttntasispi) Cheek your answers by converting your answer to EU) to iii-notation and your answer to 2(g) to M—notau'on. .... .- 30 Unit 4 (d) Given a sum-ofproduets expression. how do you expand it to a standard sum—of—pmducta (minterrnexpansion)? WM“ w‘; I wobble; “'4 and... Motown be; W'Pizr'vg hold) {e} Given a product of sums. how do you expand it to a standard product ofsums (maxtenn expansion]?madtacc, {if/(L HMS-HIV? um‘abia M3! W tum lag QM XXI, new {ocean (f) In Equatio (4—11). what theorems were used to factor f to obtain the maxterm expansion? I MED M; m vz =(K+‘Q(r+ a) (3) Why is die following expression not a maxterm expansion? f(A.B.C.D)=(A+E'+C+D)(A'+B+C')EA’+B+C+D‘) mi: firm does not malm‘n a D 0|” DJ (h) ASREllleng that there are three variables (A. B. C), identify each of the Following as a mintenn expansion. maxten'n expansion. or neither: (l) as + B'C' IUE-JTHEP- (2) {A’ + a + mm + a' + C) “Xi-5’2” {3) A + a + r: MAXTEFM (4) (A' + ma? + mm“ + C} MErTHE’F. (5) A'BC' + AB‘C + ABC (6} na'c’ “(mom tween Note that it is pDSSIhIe for a mintenn or mantterrn expansion to have only one term. {3} Given a minten'n 'LIJ remns' Ul‘ its variables. the procedure for conversion to decimal notation is (I) Replace each complemented variable with a Q and replace each urieomple— merited variable with a d. . (2) Convert the resulting binary number to decimal. (b) Convert the minterm AB’C’DE lo decimal notation. lODll—qu -—-; mm (c) Given that m” is a rnintenn of the variables A. i?r C+ D. and E. write the minterm in terms of these variables. ik. BC Dre (d) Given a maxlenn in terms of its variables. the procedure for conversion to decimal notation is (1) Replace each complemented variable with a :I and replace each uneomple— mented variable with a _-Q__.. (2) Group these 0’s and ['5 to form a binary number and convert to decimal. (e) Convert the rnmrtenn A' + B + C+ 0'4— E' to decimal notation. I D o l. I—qu—JMH (f) Given that M” is a maxterm of the variables A. B. C, D, and E. write the maxterm in terms of these variables. r [34.5 or lot -—3 KH+B'+¢+ “+50 (g) Check your answers to (b). (o). (e). and (f) by using the relation M,- = mi“. (h) Givenflo. b. c. d, e) = [1 MK). [0. 28). expressfin terms of a, b. c. d. and 2. {Your answer should contain onlyr five complemented variables.) 4: (o+b+c+d+e)(a+ 3+ {H the) {50+ 5 +— c’+d+ e) Applications of Boolean Algebra l'Mlnterm and Maxterrn Expansions 81 Study Section 4.4, General Mime rm and Mat-term Expansions. Make sure that you under- stand the notation hen: and can follow the nlge‘ortt in all of the equations. If you have difl'it:ul1'.g.r with this section. ask for help befam you take the readiness test. (a) How many different functions of four valnahlcs are possible?1 2'1, l5- 3,6 Eminamwfllezm"? 2' '- Z r £55 ‘ UnweGorl (b) Explain why than: are 22“ functions oft: variables. whld‘ can hm 0' ' (c) Write the function of Figure 4-1 in the form of Equation (443) and show that it reduces to Equation (4—3). 1r = (tn—twin) (01- Mt) (0mg [H- M3)C1+Ma)ll+Ms)£t+Mal [H to) 5 Mt: MI “3- (d) For Equ‘adli‘on (4—19), write Oul'ta'hl: indicated summations in full for the case P! = 2. +1391: 3. Eli 'bi mg = E nth; .m- brfl If pa = ooh-=er mlnlm.‘ + atlatmzmgbzms (o) Study Tables 4~3 and 4-4 carefuliy and make sun: you understand why each table entr}r is valid. Use the truth table forfandf‘ (Figure 4-1) to verify the entries in Table 4—4. If you understand the relationship between Table 4-3 and the [1'th table forfnmif’1 you should be able to perform the conversions without having to mem- orizc the table. If} Given Lhatfm. B. C) = Em“). 1.3.4. 3') The maxtcrm expansion forfis 11' M (2: 5‘! The mimenn expansion forf’ is EWCZ '5' 6) The maxlcrm expansion forf' is WM l 0t l) 3| H; (g) Work Problem 4.3 and 4.4. Study Section 4.5, Incomplele Specified Functions. {:1} State lwo reasons why some functions have “don't care" [1311119. l-W mmbnmhom mg W 055th am. w M W 41.19, 914513“: ls for El "° 4m - or comb'nnoéng? [13] Given the following table. writethe mintunn clip sion forZ in decimal form. t) 1 1 23- EMLOJQ .— 543(5ng 100 to) Write the Inaxtenn expansion in decimal form. I D 1 "Z: TTM(2,E,?)-WD(1,3,L+) MO 111 010 DDflXXOXA-m 32 Unit 4 (d) Work Problems 4.5 and 4.6. ?. Study Section 4.6. Examples uf’f‘rmh not: Construction. Finding die truth Iabiu From the problem statement is probably the most difiieult part of the process of designing a switch- ing circuit. Make sure that yOu understand how to do mis. 8. Work Problems 4.? through 4_1fll 9. Stud},r Section 4.7, Dexng originally fitment. (a) For the given parallel adder. Show the 0'5 and J‘s at the full adder (FA) inputs and out. puls when the following unsigned numbers are added: 1! + 14 = 25. Verify thatthe re— sult is correct if (345351513, is taken as :1 5-bit sum. [1‘ the sum is limited to 4 him, explain _ _ ‘ _ . , why this is an overflow tondttion. I I _. lie I l t K} m. I t t 0 {it} “‘4‘ 5 o t I o 0 tr“? 2 - S; _ i y . . ' 1-H: M mm a hmti‘d r.‘ . , - I " m b in 4- km, 1mm an t t l W“ “M m” I l D l l i l 0 LC“ (In) Review Section L4. Representntirm of Negative Ntmtbers. If we use the 2‘3 comple- ‘ 5 i l cll l merit number system to add (*5) + (—2). verify that the FA inputs and outputs are -1 1 ll it) exactly the same as in Port (Lt). However. for 2's complement. the interpretation of I) [Do t the results is quite different. After discarding C4. verify that the resultant 4-bit sum is t ’6- ' _" f' correctr and therefore no Overflow has occurred. pm H .5 (c) If we use the 1‘s complement number system to add (—5) + {—2}, Show the FA “a: ma“ inputs and outputs on the diagram below before the end—around cal-r},r is added in. Assume that Co is initially t). Then add the end—around curry (Ct) to the right— most FA, add the new earn.I (C1) into the next cell. and continue until no further changes occur. it'erifjt that the resulting sum is the correct l's complement repre— sentotion of —i‘. l E 1x0 *1. o “L b _ l t l - c [ l f.. I ‘ FA ‘9 ‘ FA 1‘4 I i | I G I I o a 1 Al -5: lot!) (t's camp) _2_: Hill {WWW-Pi U) OI ll L-——) ' mom (-4) 1D. 11. 12. ' 13. Applications of Boolean Algebra! Minternt and Maxterm Expansions 83 (a) Work the following subtraction example. As you subtract each column, place a. I over the next column if you have to borrow, otherwise place a 0. For each column, as you. compute 3:!- — y,- — by, fill in the corresponding values of bi 4 I and afl- in the truth table. If you have clone this correctly. the resulting table should match the full subtracter unth table (Table 4&6). I II I r—borrows x, mi:I 13mm!J 11000110 (—X one 0 e: 410110111 (—1" 001 I I 01 JOHGD (—diiferenee 01“ I J 011 i D '00 O I 101 D D 110 o a 111 I l (1:) Work Problems 4.11 and 4.12. Read the following and then work PmbICrn 4.13 or 4.14 as assigned: When looking at an expression to determine the required number of gates, keep in mind that the number of required gates is generally not equal to the number of AND and OR operations which appear in the expression, For example, 33+ CD+EF(G+H) centatns four hND operations and Ihrcc 0R operations. but it only requires three AND gates and two OR gates: Simulation Exercise. (Must be completed before you take the readiness test.) One purpose of this exercise is to acquaint you with the simulator that you will be using later in more complex design problems. Follow the instructions on the Unit 4 lab assignment sheet. Reread the objectives of this unit. Make sure that you understand the difference in the pro- cedures for converting mexterms and mintcnns from decimal to algebraic notation. When you are satisfied that you can meet the objectives, lake the readiness test. When you come to take the readiness test. turn in a copy of your solution to assigned simulation exercise. ...
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This note was uploaded on 01/27/2009 for the course EE 316 taught by Professor Brown during the Spring '08 term at University of Texas at Austin.

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Unit_4_SG - Study Guide in the previous units, we placed a...

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