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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 “IfB" inside the ExclusiveOR. 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) Ierepiesents 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. ContHumane! 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:: {arbIC') fa‘+t2'+c)fg"+ia+c'i i
t? = a'izg't 3.
(a) {b} e'b’eie‘bswiis‘ : ' r  .1 h i t I
EC (a'WIJEq nibHail) bctGC
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
tutnotation and in decimal notation. F: martnlrrtgﬁn.’I 2Mﬁﬁ,l,3,0r) For the same table, write the maxterrn expansion for F in iiinotation and in decimal
notation. ravine Meit1 : Ttntasispi) Cheek your answers by converting your answer to EU) to iiinotation and your answer
to 2(g) to M—notau'on. ....
. 30 Unit 4 (d) Given a sumofproduets 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 HMSHIV? 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' IUEJTHEP (2) {A’ + a + mm + a' + C) “Xi5’2”
{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 Matterm 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 41 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 44 carefuliy and make sun: you understand why each table
entr}r is valid. Use the truth table forfandf‘ (Figure 41) to verify the entries in
Table 4—4. If you understand the relationship between Table 43 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. lW 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 DDﬂXXOXAm 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 5bit 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 . .
' 1H: 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 4bit 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 calr},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 {WWWPi
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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 Spring '08
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