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Unformatted text preview: 110 Unit5 Study Guide 0.0..0IIIOI'I'DOOOII’llIO. [n this unit we will study the Kamaugh {pronounced “carn0") map. Just uboul any type of algebraic manipulation we have done so far can be facilitated by using the map. provided
the number of variables is small. I. Study Secliun 5.]. Minimum ﬁn?!” quuirchiprg Hummus.
(a) Deﬁneaminimum sum of products. . I . '
AWDﬁPWﬁﬁwwofmwpmonw ammm
Wﬁ 5mm; Met a. WNW We}. as»
Liloran w. . (b) Deﬁncamimmum product ol‘sums. ,  l .
Aproduhmt' Bis«MS {OWL DJFWGH’PWP“ W“ WWW
Wig]: Pmdudtmsmd mmwwwmbma‘ﬁ LAWs w. 2911. tm« 2. Study Section 5.2, Two and Three—Variable Kamaugia Maps. (3) P10! 1hr: given truth table on the map. Then. loop two pain: (if 1‘s on the map and write
lhe simpliﬁed form ol‘F. Q [I J F: P+§ F
New simplify F algebraically and veril‘ y that your answer is correct.
F' = ris'+ P’éi‘ + W
r 1
5 F + P55 :1 PHQ:
(b) Ha. b, c) is plotted below. Find the truLh {able for F. “Aw”...— Karnaugh Maps 111 (c) Plot [he folluwing functions on the given Kamaugh maps: FILE. 5. T) = E m0). 1. 5. 6} F103. 5. T) i TI MG. 3. 4. '1') W'h ( thctwo 1h 9m “P
)‘1Ie maps a .1 :3 11m Funcﬁmﬁ GAE. W Wm
othch (c1) Plot the following function on the given map:
JILL a) = z“ + x': + .vz Do no: make a mintcrm expansion ur a Lmth lablu before plotting. {e For a thrccvariable ma . which 5 Limes an: “ad'acenl” to s uarc 2'?
P Cl J rI QEAL_
(f ) What theorem is used when two Lenns in adjacent squares are combined?
xv+ Ky’ = X
(g) What law of Buolunn algebra justiﬁes using :I giucn l on a map in two or more
luops? X+X=K 'I ll Unit 5 (h) Each of the following solutions is nor minimum. g = rr‘ + ab la In each case. chnngu the looping on Lhc map so that the minimum solution is oblzlinonl.
(i) Work Problem 5.3. U) Find two diffcrcnl minimum sum—ol—pmducts expressions for the function G. which is
plolied below. (l[l c=£+bc+alﬁf — I {lib
5. EL
G 3. Study Section 5.3,ﬁmrVnrfczhle Karrinugh Mom. (:1) Note the locations of the minterms on 3‘ and 4vzlrinblc maps (Figures 5—30)} and
5 l0]. Memorize. this ordering This will save you a lot of Lima when you are plotting
Kamaugh maps, This ordering is valid only for lha: order ofthe variables given. If We Ia.th the maps as
shown betaW. ﬁll in the lucaiions of [he minlum‘ls: ———___.—_____. . . Karnaugh Maps 113 (b) Given the following map. write Lhe minlcrm and maidenn expansions ton” in decimal form: ab
cal I'JU EH 1  Ill‘ “0 m F=2m(15=f$Ql21I'i,l5)
I] [El ,eimﬁbgzalq at 10, n, e») 0:) Plot The fnltowing functions on the gimm maps: (I)f(w.x.y. z) = 2m(0.1.2,5,7.8.9. ID, I3. [4]
(afiwuBJH, a) = x’z' + y’z + w'xz + wyz' I
: b0 it
Your ansvwers to {1) and (2] should he [he same. w KB 5 (d) For a 4~variabie map. which squares are adjacent to Square 14? 5 ID! IEIIE'
To square 8? otq. ID; l1
(e) When we combine two adjacent J’s on a map. this con‘esPonds to applying [he Iheorem ry' + x3: = x to eliminate Ihe variable in which the two terms differ. Thus1 looping the two l's as indicated on [he following map is equivalent to combining the corresponding
mimerms algebraically: aloc’ J+ (1de2 (15351f __. a' ’c'rl' + ub'c'd _ b'c'd (The term b'c'n' can be read dime“).l from III:
map because it spans me ﬁrst and lmn columns
if"? and because i1 E5 in the second row [c'le 'l 14 Unit 5 Loop two other pairs of adjacent 1’s on this map and state the algebraic equivalent of
looping these terms. Now read the loops directly off the map and check your algebra. (f ) When we combine four adjacent 1’s on a map (either four in a line or four in a square)
this is equivalent to applying xy + xy’ = x three times: a'b’cd + a'b'cd' + ab’cd + ab'cd' = a’b’c + ab’c = b'c Loop the other four 1’s on the map and state the algebraic equivalent. (g) For each of the following maps, loop a minimum number of terms which will cover all
of the 1’s. (For each part you should have looped two groups of four 1‘s and two groups of two 1’s).
Write down the minimum sumofproduets expression for f1 and f2 from these maps. f1: a‘b+ bc‘+ a'cd+ nc'd
f2: cd+ b'c+ Ct'lod+ alo'd' (h) Why is it not possible to combine three or six minterms together rather than just two,
four, eight, etc? None at Web/\st cm be. Mini to 96W tam WWW (XY+KY’:X). Karnaugh Maps 115 (i) Note the procedure for deriving Lhe minimum pmducr of sum: from Ike map. You will
probably make fewer misiakes if you write downf' as a sum of preducls ﬁrst and [hen
complemenl il. as illuslraled by the example in Figure 5—14. (j) Work Problems 5.4 and 5.5. 4. Sludy Section 5.4. Determination of Minimum Expressions Using Exxenn'l Prime
Implicamj. (a) For the map of‘Figure 5415, list three implicanls off"ether than those which are labeled.
r:3.t:.'(.‘i'J 0160i, wad", err. (an M)
For the same map. is ae'd' a prime 1nlplicant ofF'l
Why orwhy net? NEH" a. [DH"ﬁt imﬂmwal'acﬂm Br. W'l’h I}le 'i‘ﬂ deminm 1114. d to 3,9,! (Ac). (‘0) For the given map, are any of the circled lerms Ag prime im lie nls‘? CD
Ag? Li, A c.‘ 9
Why or why not? [IIII AB'd c. am. ‘rime. iM‘ isﬁ e
cTn—mﬁcﬂgg lamina mm 0;: 13° Wmim a. Miami. i
ﬁrst, Nib, 15w mm W Pvim hm u.
m m be numbined WW". W’sW5
J“ 5% ED mat 13:. 5. Study Figure 518 carefully and [hen answer the following quesLions for [he
given map:
(:1) How many 1‘: are adjacent to molﬁ 4 (b) An: all these 1‘s covered by a single
prime implicanl'? MD (a) From your answer to (b), can you m , A r J   a.
deleime whetherB C is essential. m (cl) How many 1‘s are adjacent In mg? 2 H (e) Are all of lhese 1's covered by El. single prime imalgant? I“ {f} From your answer to (e). is B'C' cssenliai? qes
(g) How many l‘s are adjacenl to “I11? 2
m) Myisn'C¢sseneai?mg It; Mamr +9 rm an. all cowMa!
ME Crimavle [J‘n'wle :wip'ia'emi' at.
(1) Find 1 <1 other essential prime implicants and tell which minlerm 111::ch them
chemial. I I 116 Unit 5 6. (a) How do you determine if a prime implicunl is essential using a Kamaugh map? Tl allMe l‘cmdlﬁ "diamt +u MAJ :1 canine LOWE1+9
one. hawH1an lawn is maﬁal» {b} For the following map, why is A‘B‘ no! essential? all;
me, w... Combs mm 1911 at: my] Em3 M4 mgbtj
All My is HD' essential?
m cm bw new.“ “my CD
IsA‘D' essential? Why? P" I: F
MD nan/~1me (M1 AS5 BC" essential? Why?
\{eca m '5': mm
15 B'CD essenliaj? Why?
HE‘S * Due "to m “
Find the minimum sum of products,
.. .' r
F' 50+ BCH— AB‘+ E’cp
(c) Work Programmed Exercise 5. I.
((1) List all 1‘s and X's That an: adjacent to I”. lll WhyisA’C‘anessentialprimeimplicam'? AAA 1*; W K's adiaowf +13
19 w W bd A’cﬂ List all 1‘s and X's adjacent to 1'5. X?) 1”, xl‘l) KG Karnaugh Maps 117 Based on this list. wh;r can you not ﬁnd an essential prim: implicant that cnvers 1'5? m1}; MJKkadiaM in 145 W? be Wat nib?
our, [MdML Imp’a'cm Does this mean that there is no essential prime implicnnl that covers [151’ No What essential prime implicanr covers I“? ALB
Can you ﬁnd an essential prime: irnplicant that covers I”? Explain. hie — um single game, Mrwa 00% bolt 14.! KW GWLKE) Find two prime implicnnts that cover in. 66’ 3:! Give two minimum expressions for F. F: 966+ nobmtg.
: pc'c' + MIDt 8d (a) Work Problem 5.6.
[D If you have a copy of the. Logicﬂid program available. use the Karliﬁugh map [Uloriﬂl mode to help you learn tn ﬁnd minimum solutions From Kamaugh maps. This program will check ynur work an each step to make sure that you loan the terms in the
correct order, It also will check your ﬁnal answer. Work Problem 5.? using the Karnaugh map tutor. 3'. (a) In Example 4., page 9?. we derived the following function:
Z = Em{0.3,6.9) + idﬂﬂ, 1. 12,13,14115) Plot Z on the given map using X's in rcprcsent don’t care: terms. 4"! a; D’
(b) Show that the minimum sum of products is
Z = A'B‘C’D' + B'CD + AD + BCD'
Wliiciiﬁnu'dnn't care minterrns wurc assigned tin: value 1 when forming your solution? A”! dl Btd'mdig‘ UHILJ (c) Show that the minimum producL of sums forZ is z = (3+ mm“ + mm“ + mm + C+D'J(3+ C’ + D) Which one don‘t care term ufZ was assigned lhc value 1 when I'crming your solution? WI if
(it) Work Problem 5.3. Study Section 5.5. ﬁve—Variable Kai‘naugh Maps. (:1) The ﬁgure below shows :1 Three—dimensional St’ariablc map. Plul the [‘5 and loops on
the corresponding twodimensional may. and give the minimum sumuflpmducts
exprcSSion for the Function. F: REE: SUE’ﬂKBDE (b) On a S—variable map {Figure 5—2I). what are the ﬁve rrlinlcrms :Idliaccul ln
minterm 24?
Me , M25: We: Marc, mat: (:1) Work through all of the examples in this section carefully,' and mukc sure thul you
understand all of the steps. ((1) Two minimum solutions are given for Figure 524. There is a lhird minimum: sum—
of—proclucts solution. What is it? Fr— EE'C'D‘ + ErC'E + A'C‘DH—NBCDi BLDE + AC'E (c) Work Programmed Exercise 5.2. Karnaugh Maps 119 if} it?! ' 11:. (it; , raider single loaf Find the three 1's and X's adjacent to I”. Can these an be looped with a single loop?
Find the 1's and X‘s adjacent to 124. Long: the essenlial prime implicant that covers 124.
Find the 1‘s and X‘s adjacent lo 13. Limp the essential prime implieant that cot'ch 13.
Can you find an essential prime implieant that covers In? Explain. 1.2M ‘. Krgﬁzua A 6:6: 131: Cannot somlw M K13, we thfhu NOTE. I lmtuf$w+¥umm
‘ind and loop two more essential prime :mplteants. euwﬁgl, 'm i ‘ {— Find three ways to cover the remaining I on the map and give e correspon ing
minimum solutions. . . . . ICE: ﬁg
2 essmhobl WT“, WHWEC'EB ’ .H e: resignrm BEE
350MHsz F153F+Aﬂ+¥§+—+ADF +kw+ AB‘gl
F: ACE +A'ue 13* +ACE+H5e r;
(g) If you have the mfiﬁﬁd p‘r'tTg'rarn swat able—work Problem 5.9. using the Knrnaugh
map tutor. 9. Study Section 5.6. Other Uses ome'J'mugh Maps. Refer back to Figure 58 and note Ihata
' consensus term exists if there are two adjacent. but nonoveriapping prime implicants.
Observe how this principle is applied in Figure 54a 10. Work Problems 5.10. 5.11.5.1; and 5.13 When deriving the minimum solution from the
map, always write down the essential prime implicunts ﬁrst. If you do not, it is quite liker
that you will not get the minimum solution, In addition. make sure you can ﬁnd all of the
prime implieants from the map (see Problem 5,Iﬂ(b)). 11. Review the objectives and. tutu: the readiness test. ...
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 Spring '08
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