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Unformatted text preview: 54 Unit 3 Study Guide 9....'_.,...'III....I.III'.I '1. Study Section 1 l. Multiplying 0m and Factoring Expansions. (a) List lime laws or theorems which an: nSnﬁJi when mnitiplyin'gnutor factndng'expressions.
KWit‘a) : zuHKt
Ki in = (X14!) (XiF. K‘ti  ' '
(13) Use Eqﬁﬁi0:(SS§zt.0) Jot; noi‘thnfollowing: ab'c + bd = (MmKB’er)
m + mm = {abrd) gab)? r.) (c) in the following example. ﬁrst group [111: terms so. that (39.) can be applicd mo times. F1='[x~'iy‘ + $03“ + x“ + 5w +.x+ y'KW' 1;? +z')
"I..T_._..—__._._._.T "_" After applying (342). apply (33) and then ﬁnish multiplying out by using (31).
F; :2 L1+U'+E>Ln+a'+ w) k my 3+ n‘)[m'+a+ei)
:. (Isra'q L03) [ “aha 1. “IED{uﬁiﬂﬂq 32]
= 5' (mi... xlzlj ,1. 3L Hmigﬂj: DISH“91 3—3]
:: my + xh'a' + x3 + w qusmﬁn 3 r] If we did not use {32) and {333'and use only (31) on the original F; nxprossion..wn
would goncratomany more Loans: F. =[w'x‘l‘w'y’ +w'z+:rxi+xfy' +x'z+:gl+ yyi+xé§
(wi+w'x+w’y’+ivy+xy+ 391+ w:'+x_z*+y'z’}
= (w'x+ W’xy'+W'xz+‘+m'2')
W This isobviously a very inaﬁcienr'way to procéediTho moral to this story is In ﬁrst
group the terms and apply (32) and (3—3} when: possible._ I ((1) Work Programmed Exercise 3.1.Then work Pmbicm 3.6, being careful 'rmt reintroduce
any unnucessary teams in the process. (a) In Unit 2 ydu learnod how to factor a. Boolean expression. using Ibo two distributive
lawia. In addition, this unit introduced mic ofIhe theorein XY+ 17'?! = (X + ZJLX" + Y) in the factoring protons. Careful choice of the nrd'erin which mane laws and moorems
an: applied may cut down the amountof work requirod to fainteran aggression. When
factoring, it is best to apply Equation (31) ﬁrst. using as X the variable or variables
which appear most frequently. Then Equations (3—2} and (33) can beappiiod in either
order, dependi ng' on circumstanocs‘ (f) Work Progranuned Exercise 3.2. Than work Probiom 3.7. BooleanAlgebra {Continued} 55 2. Checking your artswelrs: ._
A good way to partially check your answers for oorrectttess is to substitute D‘s or 1’: for 1!;
eorne of the variables._ For catatonic, ifwe subs'tituteA = l in the .ﬁrsbend lastexpression in 
Equation (35), we get lC+DBD'+=£}BE+DUBE=[1+B+C')(1+B+D)
(1+B+E)(1+D’+E)(D+CJ
C=111w1Cc’ Sioﬁhﬂyrsuhstitutingzt = 0,3 = D we get 0 + 0 + o + one = (a + C‘)(0.+ ma} + EXIT + Eltl + C)
= one .I Verify that the result is also correct whenA = U and B = l. 3. The method which you use to go your :inswor is very importtmt in this unit. Ifit takes
you two pages of algebra and one hour of timeto Work a problem which centre solvedin .' _
10 minutes with three lines of work. you have not learned the material in this unit! Ewen '': .
if you get the correct answenyour wart: is not satisfactory if you worked the problem by '
an excessively long and littleconsuming method. It is important that you learn Luz"solve
simple problems in a simple meon.er—otl:terwise._when you are {asked to some a complex. '
problem, you will getbogged down and never get the answer. When you are given a prob
lent to solve. do notjust plunge in, but ﬁrst askyourseti'. "What is the easiestwey to work
this problem?“ For example. when you are askedto multiply out an expression, do not just 
multiply it out by brute force, term by term. Instead, est: yottrseli’,_"I.low can I group the I
toms and which theorems shouldl apply ﬁrst in order to. rotiu'ee theamount of w'cirlt?’g 
(See Study Guide Part 1.) After you haveworked out Problemsaﬁ end 3_.?.cot_ripare your __ . _ .
solutions with those in the solution book. If your solution required substantially mere. '. ' 3 7‘
work than the one ‘m‘the solution book. rework the problem and try to get the answeriu a '1
more straightforward manner. ' 4. Study Section 3.2. ExclusiveOR and'eqtjvaJence Opemﬁm. (a) Prbve Theorems (3—3} through.(313}. You should be a e to prove'these oath aige ‘
braieally and by using strut}: Meg—[3); L115: {35.69 3 '3 :US‘H‘X‘l'b i " rtEBO: towo. we we“ emmetteeth'
(3431 if“ {:1 [3,,” : i'tr . r :txtvttHDet(xltm'kﬁ
{3’53} _; a" ' (5’53": 3‘, + “I = a'trhuxﬁi vita Mel
' “‘1‘ the: tastier}: to? TatHz") ,
KM= “It M = vex —' x'l‘tbtW + H rim?
£3va = D : x‘tt‘trttt'ir otwwcr‘ﬁJ (b) Show that (1'?! + 1'y3' = .1)! + x'y'. Memorize thieresuit .5 $11!; .1. lLl‘fzt'f' H31“?
(xa‘ﬂ'gf r: [x'waﬂxttaf) : fabK. ”as
(c) Px'ove 1116013!!! (3?5). I 43; I  1® t
was}: LKWU')’ W *a’ 3" “‘3 H “‘ ‘3 ' 53
' Lx'J'Etr x'at (”’33 1] ti 55 Unit 3 {d18howthat(x=ﬂ)=x’.(xax)=l andttay)’={x=y’).
(NED)=sro+r'I (K: r): “:ij (i=3? =[WHIJ
3:: = ﬁHt
" K +X
(e) ExpreschEyju’itt m’tertnsgof‘exc‘iusiveokh .: x;3*}‘+1xl1l:kxgwr}
©9511’51KkitIt1): K'jt XL; :éteﬁ] (f1 Work Problems3.8 and 3.9. 5. Stud}.r Section '13, The Consensus Theorem The consensus theorem is an important method
for simpiifying switching fnncn'ons.
(a) In each of the following expressionsrﬁnd the consensus tern: and eliminate it: abc'd +31»: +32%;
(a' + b + emu +'d)(E‘+£==.dj_
non: + a'bq + out“ + Nag
t._—.’ (b) Eliminate two terms from the following expression by applying [netconsensus
theorem: 'A’B’G + new +wco+ ABID' + BFo + new: (Hint: First compare the ﬁrst term with each of the remaining terms to see if a
consensus exists, then compete the second term with each of the. remaining
terms; etc. 1 to) Study the example given 111 Equations (32213nd (3A231eareﬁilly. Now let us start with
the fourterm form of the expression (Equation 3 22): A’C'D + A'BD + ABC + ACD' Can this he reduced directiy to three terms. by the application of the consensus
theorem? Before we can reduce this expression. we must add another term Which term
eanbe added by applying the consensus theorem? '
. NC) 'A at), A on gives eco
Add this term, and then reduce the expression to three terms. After this reductionrcan
the term which was added he removed? Why not? A c D +¥BQ+ keg Aco +sw= A'C‘DiACD '+ac:b 5 math
l_t_—.“—.L———.II €013th WOW. w— m aw “We9 (d) Eliminate two terms from the following expression by'applying 111%3] consensus
theorem: 1—~———r———1
(a' + c' + and + b + cite + b + ”W Mid—)— Use brackets to indicate how you formed the consensus terms. (Hint: First, ﬁnd the
consensus of the ﬁrst two remand eliminate it.) Boolean Algebra (Continued) 57 (e) Derive Theorem (3—3) by using the consensustheoreut; t
(“3.1 cm): M's set» Katg! = W : ””5
(1') Work Programmed Exercise 3.3. Than woritProblem.3.10_L 5. Study Section 3.4. Algebraic Simplﬁ‘icatfdn ' afﬁ'wr‘rcltt'ng Expressions.
(a) What themems are used for:
Combining terms? x3 _+. k .11: X Eliminating terms? _ I
‘tttttgslt; xt’xtgpmc J K3 + w‘ergen x3+ Ice
Eliminating IiteruJS?
K+ ‘53 = K +‘j
Adding redundant terms? I t
336:0; KHCISJ'; “3+.ng KiriKi“? 33' I
Factoring or multiplying out? _
ﬂy 1* e}; :5 + m— (Mumt’w) ﬁrst) six wow (WIDE)...
03') Note that in the example of Equation (327), the redundant term W“ was added and 
then was eliminated later after it had been used to'elitninute another term) Why was it  '
possible to eliminate WZ‘ in this example? ‘ .
The. "lawns. tot: ml, 345' arm. Si‘l ll u". Hate. “Premium
:9 The. wHSEWtS‘Iru tamt wa’ can be eh'mn‘nnlei
If a term has been. added by theconsensus theorem. it may not always bepos'sible
to eliminate the terntimer by the consensus theorem. Why? The "hermi Ltasgltn add +lne. wmus new wanu hm (c) You will need connect—able practice to develop skill in simpliiying switching expressions.
Work through Freermed Exercises 3.4 and3.5.
(d) Work Problem 311.
j{e) When simplifying .an expression using Bool'een algebra, two frequaritly asked
questionsare. .
(I) Where do} begin? m
(2) new do I new when I and ﬁnished? 
1n sniwet to (1). it is generally best to tn simple techniques such as ccmbintng‘tenns or
eliminating terms and literals before'up'ng more complicated things suchas using the
consensus theoremor adding redundant terms. Question [2) is generally difﬁcult to answer
because it may be impns'si ble to simplify some expressions without ﬁrst adding redundant
terms. We will usually tell you how many terms to expect in thentinimuzn solution an
that. you will not have to waste time trying to simplify an expression which is already
minitrtized‘ In Units 5 and 6, you will team systematic techniques which will guarantee
ﬁnding the minimum solution. 7. Study Section 3.5. Proving infinity cgl'mt Equation. [3) When attempting to prove that an equation is valid. is it— pennissibie'to add the same
expression to both side's? Explain. bio—"ﬁne 5W “preteen wtwa be. valid we“ Hi
4m: (Rising wristIsiah is mini" 58 Unit3 (b) Work Problem 3.12.
(c) Show that (333) and (334) are true by considering both x = 0 and x = 1. For (3 3'3): IF X: o x+y= x+2 valuesio 3:2 wktok “fruit 3:2,
IF x=l X‘Ij: x+z Veiuaesép "I)ukl€1\l$ Riwdsyr'frlﬁ. For C3”3‘f)3 IF X=0) KY: X2 YEJdLaesEa 0:0, which” QIWQJ; frag
IF x=t, xv: x2. reatucer wéo y=z, 0351:le {me up 3:2; (d) Given that a'(b + d’) = a’(b + e’), the following “proof" shows that d = e:
Lu:— «=1, 1,: o, d: a, 2:1 a'(b+d') =a'<b + e') LHQ’: O (0 +0 :0 a+b’d:a+l:e/<: cannot Cancel q,
Riff? (9 [0+o3= O —_: LHS b’d=b’e ‘\ t
but (1% e d=e é!“ Cunno Comte! b State two things that are wrong with the “proof.” Give a set of values for a, b, d, and e
that demonstrates that the result is incorrect. 8. Reread the objectives of this unit. When you take the readiness test, you will be expected
to know from memory the laws and theorems listed at the end of Unit 2. Where
appropriate, you should know them “forward and backward”; that is, given either side of
the equation, you should be able to supply the other. Test yourself to see if you can do
this. When you are satisﬁed that you can meet the objectives, take the readiness test. Boolean Algebra (Continued) In this unit we continue our study of Boolean algebra to learn additional methods for
manipulating Boolean expressions. We introduce another theorem for multiplying out and
factoring that facilitates conversion between sumof—products and product—of—sums expres
sions. These algebraic manipulations allow us to realize a switching function in a variety of
forms. The exclusiveOR and equivalence operations are introduced along with examples
of their use. The consensus theorem provides a useful method for simplifying an expres
sion. Then methods for algebraic simpliﬁcation are reviewed and summarized. The unit
concludes with methods for proving the validity of an equation. 3.1 Multiplying Out and Factoring Expressions Given an expression in productof—sums form, the corresponding sumof—products
expression can be obtained by multiplying out, using the two distributive laws: X(Y+Z) =XY+XZ (31)
(X + Y)(X + Z) = x + YZ (3—2) ...
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 Spring '08
 Brown

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