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Unit_3_SG - 54 Unit 3 Study Guide 9'III.I.III.I'1 Study...

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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. KWi-t‘a) : zuH-Kt K-i- in = (X14!) (Xi-F. K‘t-i- - ' ' (13) Use Eqfifii0:(S-S§zt.0) Jot; noi‘thnfollowing: ab'c + bd = (MmKB’er) m + mm = {ab-rd) gab)? r.) (c) in the following example. first group [111: terms so. that (3-9.) can be applicd mo times.- F1='[x-~'i-y‘ + $03“ + x“ + 5w +.x+ y'KW' 1;? +z') "I..T_._..—__._._._.T "_" After applying (342). apply (3-3) and then finish multiplying out by using (3-1). F; :2 L1+U'+E>Ln+a'+ w) k my 3+ n‘)[m'+a+ei) :. (Isra'q- L03) [ “aha 1. “IED-{ufiiflflq 3-2] = 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 {3-33'and use only (3-1) 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 (3-2) 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 of-Ihe 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 amount-of work requirod to fainter-an aggression. When factoring, it is best to apply Equation (3-1) first. using as X the variable or variables which appear most frequently. Then Equations (3—2} and (3-3) can be-appiiod in either order, dependi ng' on circumstanocs‘ (f) Work Progranuned Exercise 3.2. Than work Probiom 3.7. Boolean-Algebra {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 last-expression in - Equation (3-5), we get l-C+D-BD'+=£}-BE+D-UBE=[1+B+C')(1+B-+D) -(1+B+E)(1+D’+E)(D+CJ C=1-1-1w1-Cc’ 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 when-A = 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 little-consuming 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 not-just plunge in, but first askyourseti'. "What is the easiestwey to work this problem?“ For example. when you are asked-to 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 the-amount of w'cirlt?’g - (See Study Guide Part 1.) After you have-worked 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. Exclusive-OR and'eqtjvaJence Opemfim. (a) Prbve Theorems (3—3} through.(3-13}. 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: tow-o. we we“ emmetteeth' (3431 if“ {:1 [3,,” : i'tr . r :txtvttHDet-(xltm'kfi {3’53} _; a" ' (5’53": 3‘, + “I = a'trh-u-xfii- vita Mel -' “‘1‘ the: tastier}: to? Tat-Hz") , KM= “It M = vex —' x'l‘tbt-W + H rim? £3va = D : x‘tt‘trttt'i-r otwwcr‘fi-J (b) Show that (1'?! + 1'y3' = .1)! + x'y'. Memorize thieresuit .5 $11!; .1. lLl‘fzt'f' H31“?- (xa‘fl'gf r: [x'waflxt-taf) : fab-K. ”as (c) Px'ove 1116013!!! (3-?5). I 43; I | 1® t was}: LKWU')’ W *a’ 3" “‘3 H “‘ ‘3 ' 53 ' Lx'J'E-t-r x'at (”’33 1] ti 55 Unit 3 {d18howthat(x=fl)=x’.(xax)=l andttay)’={x=y’). (NED)=-s-ro+r'-I (K: r): “:ij (i=3? =[WHIJ 3:: = fi-Ht " K +X (e) ExpreschEyju’itt m’tertnsgof‘exc‘iusiveokh .: x;3*}‘+1xl1l:kxgwr} ©9511’51Kkit-It1): K'j-t- XL; :étefi] (f1 Work Problems-3.8 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 [net-consensus theorem: 'A’B’G + new +wco+ ABID' + BF-o + 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 (3-2213nd (3A231earefiilly. Now let us start with the four-term 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‘Di-ACD '+ac:b 5 math l_t_—.“—.L———.II €013th WOW. w— m aw “We-9 (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 consensus-theoreut; t (“3.1 cm): M's set» Kat-g! = 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 ‘tt-t-ttgslt; xt’xtgpmc J K3 + w‘ergen x3+ Ice Eliminating IiteruJS? K+ ‘53 = K +‘j Adding redundant terms? I t 336:0; K-HCISJ'; “3+.ng Kiri-Ki“? 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 (3-27), 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‘I-ru- tam-t wa’ can be eh'mn‘nnlei If a term has been. added by theconsensus theorem. it may not always bepos'sible to eliminate the tern-timer by the consensus theorem. Why? The "hermi- Ltasgl-t-n add +lne. wmus new wan-u hm (c) You will need connect—able practice to develop skill in simpliiying switching expressions. Work through Freer-med 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-'u-p'ng more complicated things such-as using the consensus theorem-or 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 then-tinimuzn 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 wrist-Isiah is mini" 58 Unit3 (b) Work Problem 3.12. (c) Show that (3-33) and (3-34) 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‘I-j: x+z Veiuaesép "I)ukl€1\l$ Riwdsyr'frlfi. For C3”3‘f)3 IF X=0) KY: X2 YEJd-Laes-Ea 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 sum-of—products and product—of—sums expres- sions. These algebraic manipulations allow us to realize a switching function in a variety of forms. The exclusive-OR 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 product-of—sums form, the corresponding sum-of—products expression can be obtained by multiplying out, using the two distributive laws: X(Y+Z) =XY+XZ (3-1) (X + Y)(X + Z) = x + YZ (3—2) ...
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