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logic functions and equations

Logic Functions And Equations Examples And Exercises 1st Edition Bernd Steinbach ,Christian Posthoff - Solutions

1. University administrators know well the benefi ts that follow notable success in college sports: increased applications for admissions, increased income from licensed logo merchandise, more lucrative television deals, post-season game revenue and more successful alumni fund drives. Th e idea
30. We say that an end pursued in its own right is more complete than an end pursued because of something else, and that an end that is never choiceworthy because of something else is more complete than ends that are choiceworthy both in their own right and because of this end. Hence, an end that
29. Th ough it is possible that REM sleep and dreaming are not necessary in the adult, REM deprivation studies seem to suggest otherwise. Why would REM pressure increase with deprivation if the system is unimportant in the adult?Each of the following passages contains a single argument. Using the
★28. Anyone familiar with our prison system knows that there are some inmates who behave little better than brute beasts. But the very fact that these prisoners exist is a telling argument against the effi cacy of capital punishment as a deterrent. If the death penalty had been truly eff ective
27. Since the secondary light [from the moon] does not inherently belong to the 1 moon and is not received from any star or from the sun, and since in the whole universe there is no other body left but the earth, what must we conclude?What is to be proposed? Surely we must assert that the lunar
26. Not only is the sky blue [as a result of scattering], but light coming from it is also partially polarized. You can readily observe this by placing a piece of Polaroid (for example, one lens of a pair of Polaroid sunglasses) in front of your eye and rotating it as you look at the sky on a clear
★25. Contrary to the tales of some scuba divers, the toothy, gaping grin on the mouth of an approaching shark is not necessarily anticipatory. It is generally accepted that by constantly swimming with its mouth open, the shark is simply avoiding suff ocation. Th is assures a continuous fl ow of
24. Many people believe that a dark tan is attractive and a sign of good health, but mounting evidence indicates that too much sun can lead to health problems.One of the most noticeable eff ects is premature aging of the skin. Th e sun also contributes to certain types of cataracts, and, what is
23. If a piece of information is not “job relevant,” then the employer is not entitled qua employer to know it. Consequently, since sexual practices, political beliefs, associational activities, etc., are not part of the description of most jobs, that is, since they do not directly aff ect
★22. Th e stakes in whistleblowing are high. Take the nurse who alleges that physicians enrich themselves in her hospital through unnecessary surgery; the engineer who discloses safety defects in the braking systems of a fl eet of new rapid-transit vehicles; the Defense Department offi cial who
21. Neither a borrower nor lender be For loan oft loses both itself and friend, And borrowing dulls the edge of husbandry.Each of the following passages contains a single argument. Using the letters “P”and “C,” identify the premises and conclusion of each argument, writing premises fi rst
20. Corn is an annual crop. Butcher’s meat, a crop which requires four or fi ve years to grow. As an acre of land, therefore, will produce a much smaller quantity of the one species of food than the other, the inferiority of the quantity must be compensated by the superiority of the price.Each of
★19. Poverty off ers numerous benefi ts to the nonpoor. Antipoverty programs provide jobs for middle-class professionals in social work, penology, and public health. Such workers’ future advancement is tied to the continued growth of bureaucracies dependent on the existence of poverty. Each of
18. Every art and every inquiry, and similarly every action and pursuit, is thought to aim at some good; and for this reason the good has rightly been declared to be that at which all things aim.Each of the following passages contains a single argument. Using the letters “P”and “C,”
17. An ant releases a chemical when it dies, and its fellows then carry it away to the compost heap. Apparently the communication is highly effective; a healthy ant painted with the death chemical will be dragged to the funeral heap again and again.Each of the following passages contains a single
★16. Radioactive fallout isn’t the only concern in the aft ermath of nuclear explosions.Th e nations of planet Earth have acquired nuclear weapons with an explosive power equal to more than a million Hiroshima bombs. Studies suggest that explosion of only half these weapons would produce enough
15. Women of the working class, especially wage workers, should not have more than two children at most. Th e average working man can support no more and the average working woman can take care of no more in decent fashion.Each of the following passages contains a single argument. Using the letters
14. To every existing thing God wills some good. Hence, since to love any thing is nothing else than to will good to that thing, it is manifest that God loves everything that exists.Each of the following passages contains a single argument. Using the letters “P”and “C,” identify the
★13. Since private property helps people defi ne themselves, since it frees people from mundane cares of daily subsistence, and since it is fi nite, no individual should accumulate so much property that others are prevented from accumulating the necessities of life.Each of the following passages
12. Cats can think circles around dogs! My cat regularly used to close and lock the 1 door to my neighbor’s doghouse, trapping their sleeping Doberman inside.Try telling a cat what to do, or putting a leash on him—he’ll glare at you and say, “I don’t think so. You should have gotten a
11. Profi t serves a very crucial function in a free enterprise economy, such as our own. High profi ts are the signal that consumers want more of the output of the industry. High profi ts provide the incentive for fi rms to expand output and for more fi rms to enter the industry in the long run.
★10. Punishment, when speedy and specific, may suppress undesirable behavior, but it cannot teach or encourage desirable alternatives. Th erefore, it is crucial to use positive techniques to model and reinforce appropriate behavior that the person can use in place of the unacceptable response
9. An agreement cannot bind unless both parties to the agreement know what they are doing and freely choose to do it. This implies that the seller who intends to enter a contract with a customer has a duty to disclose exactly what the customer is buying and what the terms of the sale are.Each of
8. Th e classroom teacher is crucial to the development and academic success of the average student, and administrators simply are ancillary to this eff ort.For this reason, classroom teachers ought to be paid at least the equivalent of administrators at all levels, including the
★7. It really does matter if you get enough sleep. We need sleep to think clearly, react quickly, and create memories. Studies show that people who are taught mentally challenging tasks do better aft er a good night’s sleep. Other research suggests that sleep is needed for creative problem
6. Th e fact that there was never a land bridge between Australia and mainland Asia is evidenced by the fact that the animal species in the two areas are very diff erent. Asian placental mammals and Australian marsupial mammals have not been in contact in the last several million years.Each of the
5. Artists and poets look at the world and seek relationships and order. But they translate their ideas to canvas, or to marble, or into poetic images. Scientists try to fi nd relationships between diff erent objects and events. To express the order they fi nd, they create hypotheses and theories.
4. When individuals voluntarily abandon property, they forfeit any expectation of privacy in it that they might have had. Th erefore, a warrantless search or seizure of abandoned property is not unreasonable under the Fourth Amendment.Each of the following passages contains a single argument. Using
3. As the denial or perversion of justice by the sentences of courts, as well as in any other manner, is with reason classed among the just causes of war, it will follow that the federal judiciary ought to have cognizance of all causes in which the citizens of other countries are concerned.Each of
2. Since the good, according to Plato, is that which furthers a person’s real interests, it follows that in any given case when the good is known, men will seek it.Each of the following passages contains a single argument. Using the letters “P”and “C,” identify the premises and conclusion
1. Titanium combines readily with oxygen, nitrogen, and hydrogen, all of which have an adverse eff ect on its mechanical properties. As a result, titanium must be processed in their absence.Each of the following passages contains a single argument. Using the letters “P”and “C,” identify the
4 Apply vectorial derivatives in order to verify whether given functions f1(x) and f2(x) are self-dual functions A logic function is self-dual if and only ifProve this theorem, identify the number of self-dual functions of n variables, and verify whether given functions f1(x) (4.18) and f2(x)
3 Prepare the functions f1(x) and f2(x) as objects number 1 and 2, and the VT x1, x2, x3, x4 as object number 3.A logic function is self-dual if and only ifProve this theorem, identify the number of self-dual functions of n variables, and verify whether given functions f1(x) (4.18) and f2(x) (4.19)
2 How much self-dual functions of n variables exist?A logic function is self-dual if and only ifProve this theorem, identify the number of self-dual functions of n variables, and verify whether given functions f1(x) (4.18) and f2(x) (4.19) are selfdual functions af (x) Ox = 1.
1 Prove the theorem.A logic function is self-dual if and only ifProve this theorem, identify the number of self-dual functions of n variables, and verify whether given functions f1(x) (4.18) and f2(x) (4.19) are selfdual functions af (x) Ox = 1.
4 Calculate the stable states using an intersection and an m-fold maximum.Calculate the stable states of the asynchronous finite-state machine given by the following TVL in ODA-form Y $1 82 83 ds1 ds2 ds3 0 0 - 0 0 0 0 0 1 1 0 0 10 1 1 0 0 1 1 1 - 01 0 0 1 F(x, y, s, ds) = 0 10 10 0 1 1 1 1 0 1 0 1
3 Calculate F(s, ds) = max2(x,y) F(x, y, s, ds) as object number 4.Calculate the stable states of the asynchronous finite-state machine given by the following TVL in ODA-form Y $1 82 83 ds1 ds2 ds3 0 0 - 0 0 0 0 0 1 1 0 0 10 1 1 0 0 1 1 1 - 01 0 0 1 F(x, y, s, ds) = 0 10 10 0 1 1 1 1 0 1 0 1 1 1 10
2 Prepare a VT x, y as object number 2 and solve the equation ds1 ds2 ds3 = 1 and store the result as object number 3.Calculate the stable states of the asynchronous finite-state machine given by the following TVL in ODA-form Y $1 82 83 ds1 ds2 ds3 0 0 - 0 0 0 0 0 1 1 0 0 10 1 1 0 0 1 1 1 - 01 0 0
1 Store the given TVL as object number 1.Calculate the stable states of the asynchronous finite-state machine given by the following TVL in ODA-form Y $1 82 83 ds1 ds2 ds3 0 0 - 0 0 0 0 0 1 1 0 0 10 1 1 0 0 1 1 1 - 01 0 0 1 F(x, y, s, ds) = 0 10 10 0 1 1 1 1 0 1 0 1 1 1 10 1 1 1 0 1 0 1 1 000 0 0 0
Calculate all three simple derivatives of the functionwith regard to x4 using both types of possible XBOOLE operations, visualize the Karnaugh maps, and verify the relations between these three simple derivatives f(x1, x2, x3, x4) = (x1T2 T2x324) V1x24
12 Verify the inequalitiesusing the objects number 27, 23, 1, 33, and 37, respectively. Mins,) f(x) < Minrs f(x), (4.3) Mines f(x) f(x), (4.4) f(x) Maxrs f(x), (4.5) Maxrs f(x) < Min r3,14) f(x) (23,24) (4.6)
Calculate all three partial differential operations of the functionwith regard to (x3, x4), draw their graphs, and verify the relations between these three partial differential operations f(x1, x2, x3, x4)=(212) x3 Vx1x4 V (x1 x2) (x3 V x4)
2 Prepare the XBOOLE-monitor in such a way that the TVL of Gd(a, x, dx)is the object number 1, and execute the PRP e42 gds.prp.Transform the differential representation Gd(a, x, dx) of the graph calculated in Exercise 4.1 into its sequential representation Gs(a, x, xf).Use the equation xfi = xi ⊕
Exercise 4.1 (Graph Equation). Solve the graph equation F(a, x, dx) =G(a, x, dx) whereand Exp xp xp xxxp xp xp xxx Exp xxxx Exp xp xpx\ xxx Exp 1x xxx / Exp xxxx Exp xp xxxp xp lap xxx Exp xp xxxx xxx xp xp xxx xxxx Exp xp xx ^ Exp xp xxx / xp xp xpxx=(xp'x')
Find an antivalence form for the following function 3 f3 ((x1 x2x3)(2243) 214) V1. ==
Find an antivalence form for the following function 2 f2 ((x1 Vx23x4)((T2 V x4)x134) V 2x3) V (1 V 24);
Find an antivalence form for the following function 1 fi (x Vyz)(x V 2);
Find a disjunctive and a conjunctive form for the following function: 3 f3 = ((x12x3)(x2x4x3)x14) V1.
Find a disjunctive and a conjunctive form for the following function: 2 f =((x1 V 234)((T2 V x4)x134) V 2x3) V (1 V 24);
Find a disjunctive and a conjunctive form for the following function: 1 fi (x Vyz)(x V 2);
(NAND and NOR).2 Compare f(xy) 2 = (xVy) V2 with g = x(y2) = I CV (y v 2).
(NAND and NOR).1. Compare the two functions f = (xy) = (xy) Az and g = |(y|2) = x^(y^2).
Can the following rules be used?The composition of functions is a powerful mechanism that can and will be used very often. The basic idea is the implementation of “smaller”functions and its combination by other functions. As an illustration answer the following question 1 x V (y2) = (x Vy) ~(x
Are the following pairs of formulas equivalent– try to prove this equivalence by building the disjunctive normal form of the two formulas. 1 fi = (VyV2) (Vy)(V2); f = ~ 2; (xy)2; f2=x(y 2); 2 fi 3 fi = [(xy) (xvy)][(y)(xy)]; f = x | y; 4 f1 = (xy) V (x2)y; f2 = (xy) (y xz).
Which one of the following formulas defines a tautology? >> 1 (xy) ((x V2) (y V2)); 2 ((xy) 2)(xyz);
Transform the following functions into the respective arithmetic polynomials: 1 f12x3; 2 f (12) 23; = 3 f 12223 VT1T2T3. =
Find the antivalence polynomial and the equivalence polynomial for the following functions: 1 f (12) 23: = 2 f(x1 = x2(x2 x3)); 3 f((12) VT3)|1.
Generalize the two last items of the previous question to larger values of n: 1 f(x1 V2 VVn)(T1 VT2 VV In); = 2 f(x1 V2 V3)(T1 VT2 VT3) 24 =
Find the conjunctive and the equivalence normal forms of the functions given 1 f ((x1 Vx23x4)((T2 V 4) 1374) V 223) (T1 V 24); = 2 f((x12x3)(x2x4x3)x14); = 3 f(x1 V2 V3 VV9 V10) (T1 V2 V3 VVT9 VT10); 4 f(x1 V2 V3)(T1 V2 V3) 242526272829210.
Find the disjunctive and the antivalence normal form of the following functions: 1 f((x1 V x23x4)((T2 V 4) 2134) V 223) (T1 V 24); = = 2 f((x12x3)(x2x4x3)x14) V1; 3 f(x1 V2 V3 VV 29 V 10) (T1 V2 V3 VVT9 VT10); 4 f(x1 V2 V3) (T1 VT2 VT3) 24 25 26 27 28 29 10. =
Which functions are defined by the following formulas (equations):Give the disjunctive, conjunctive, antivalence and equivalence normal forms for these functions. 1 (xy) ((y2) (2x)); 2 (Vy) V (az) | (x~y); 3T (~(x2)); 4(((y)) | y) | 2.
1 How many functions exist with f(x1,...,xn) = f(x1,...,xn)?
Answer the same question for ∧, ∨, ∼, →. fo(x) = f(x, y) = xy, f(x, y) = x^y, f2(x,y) = xvy, f4(x, y) = x~y, f(x, y) = x y, y)=xy, xy, f6(x, y) = xy=x^y, f(x,y) = x | y=x Vy
Show that the expressions (x ⊕ y) ⊕ z = 1 and x ⊕ (y ⊕ z) = 1 define the same function? fo(x) = f(x, y) = xy, f(x, y) = x^y, f2(x,y) = xvy, f4(x, y) = x~y, f(x, y) = x y, y)=xy, xy, f6(x, y) = xy=x^y, f(x,y) = x | y=x Vy
Solve Boolean Equations fo(x)=, fi(x, y) = x^y, f2(x, y) = x Vy, f3(x, y) = xy, f4(x, y)=xy, f(x, y) = xy, fe(x, y) = xy=x^y, fr(x,y) = xy=xvy
1 Define the functionsusing TVLs. fo(x)=x, f(x, y) = x^y, f2(x, y) = xvy, f3(x, y)=xy, f4(x,y) =~y, f(x, y) = xy, f6(x, y) = xy=x^y, f(x,y) = x | y=xvy
Let x and y be two elements of Bn. Show that 3 ||xy|| = ||x|| + ||y|| - ||x Vy||.
Let x and y be two elements of Bn. Show that 2 ||x vy|| = ||x||+|y||-||xy||;
Let x and y be two elements of Bn. Show that 1 ||X|| = n - ||x||;
Let be given two vectorsFind all vectors z with x, y B" with x
4 Let be given the vector x = (10010101) ∈ B8. (a) Find all y = B8 with x
3 Find the binary vectors x with 2n-1 dec(x) < 2.
Find the vector Find the vector x for dec(x) = 19 in B8. xB6 with dec(x) = 19.
Find the decimal equivalent dec(x)for the vectors (1001) B4, (01101) B5, (110010) B6. E
Show that is equivalent to xyz
Show that (y v2) if x y or x 2.
Show that if 21 y1 and x2 y2 then 12 y192 and (21 V 2)
Show that < (Vy) and x > xy.
9 Verify whether these function solve the associated equation. Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and verify them:x1x3y ∨ x2(x3 ⊕ y) ∨ x1x3y = 1, (4.43)x1x3y ∨ x2(x3 ⊕ y) ∨ x2y · (x1 ∨ x3) = 1,
8 Calculate all functions of the solution sets based on the created mark functions q(x) and r(x). Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and verify them:x1x3y ∨ x2(x3 ⊕ y) ∨ x1x3y = 1, (4.43)x1x3y ∨ x2(x3 ⊕
7 Calculate the mark functions q(x) and r(x) of the solution sets for all equations that are solvable with regard to y, but not uniquely. Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and verify them:x1x3y ∨ x2(x3 ⊕ y)
6 Verify the solution of the equations which are unique with regard to y. Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and verify them:x1x3y ∨ x2(x3 ⊕ y) ∨ x1x3y = 1, (4.43)x1x3y ∨ x2(x3 ⊕ y) ∨ x2y · (x1 ∨
5 Calculate the solution functions for all equations that are unique with regard to y. Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and verify them:x1x3y ∨ x2(x3 ⊕ y) ∨ x1x3y = 1, (4.43)x1x3y ∨ x2(x3 ⊕ y) ∨ x2y
4 Which equation is solvable with regard to y? Evaluate the calculated Karnaugh maps for this equation. Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and verify them:x1x3y ∨ x2(x3 ⊕ y) ∨ x1x3y = 1, (4.43)x1x3y ∨
3 Which equation is unique with regard to y? Evaluate the calculated Karnaugh maps for this equation. Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and verify them:x1x3y ∨ x2(x3 ⊕ y) ∨ x1x3y = 1, (4.43)x1x3y ∨ x2(x3
2 Which equation is unique with regard to y? Evaluate the calculated Karnaugh maps for this equation. Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and verify them:x1x3y ∨ x2(x3 ⊕ y) ∨ x1x3y = 1, (4.43)x1x3y ∨ x2(x3
1 Prepare the functions of the left-hand sides of (4.43), (4.44), (4.45),(4.46) as objects number 1, 2, 3, and 4, respectively. Assign the variable y to object number 4. Check the following equations for uniqueness and solvability with regard to y. Calculate all solution functions y = g(x) and
In expression (4.29) of f(x) it is not directly visible whether the function is linear with regard to certain variables:f(x) = x1(x3 ⊕ x2x4) ∨ x1(x2 ⊕ x3)x4 ∨ x1 x3 x4. (4.29)Check by means of (4.28) whether function (4.29) is linear, separately for each variable of the set {(x1, x2, x3,
In the expression (4.26) of f(x)each variable appears both negated and not negated:f(x) = x2x3x4 ∨ x4(x1 ∨ x2) ∨ x1x3(x2 ⊕ x4). (4.26)Check by means of (4.24) and (4.25) whether function (4.26) is monotonously increasing or monotonously decreasing, separately with regard to each variable of
4 Simplify the three functions as much as possible and verify the result Check on which variables the given functions f1(x), f2(x), and f3(x) depend:f1(x) = (x1x2 ⊕ x3x4) ∨ x3x4 ∨ x1x2 ∨ x1x3, (4.13)f2(x) = x2x4 ∨ x1 · ((x3 ⊕ x2) ∨ (x2 · x3)), (4.14)f3(x) = x1x2 ∨ x3x4 ∨ (x1 ⊕
3 Calculate the simple derivatives of each function (4.13), (4.14), and(4.15) with regard to each variable. Are there independences?Check on which variables the given functions f1(x), f2(x), and f3(x) depend:f1(x) = (x1x2 ⊕ x3x4) ∨ x3x4 ∨ x1x2 ∨ x1x3, (4.13)f2(x) = x2x4 ∨ x1 · ((x3 ⊕
2 Prepare VTs x1, x2, x3, and x4 as objects number 4, 5, 6, and 7.Check on which variables the given functions f1(x), f2(x), and f3(x) depend:f1(x) = (x1x2 ⊕ x3x4) ∨ x3x4 ∨ x1x2 ∨ x1x3, (4.13)f2(x) = x2x4 ∨ x1 · ((x3 ⊕ x2) ∨ (x2 · x3)), (4.14)f3(x) = x1x2 ∨ x3x4 ∨ (x1
1 Prepare the functions f1(x) (4.13), f2(x) (4.14), and f3(x) (4.15) as objects number 1, 2, and 3, respectively.Check on which variables the given functions f1(x), f2(x), and f3(x) depend:f1(x) = (x1x2 ⊕ x3x4) ∨ x3x4 ∨ x1x2 ∨ x1x3, (4.13)f2(x) = x2x4 ∨ x1 · ((x3 ⊕ x2) ∨ (x2 · x3)),
Exercise 4.18 (Static Functional Hazard). Calculate all static functional hazards of function (4.2) restricted to all possible changes of values of x3 and x4. Practical tasks:1 Reload the TVL-system of Exercise 4.7, minimize object number 41 that represents ϑ(x3,x4)f(x) and write down this TVL.2
Calculate all possible critical transitions of the asynchronous finite-state machine given in Exercise 4.16. Practical tasks:1 Reload the TVL-system of Exercise 4.16 as the basis of all further calculations.2 Solve the equation ds1 ds2 ∨ds1 ds3 ∨ds2 ds3 = 1 as object number 10.3 Find the
3 Execute this PRP and evaluate the results.
2 Write a PRP that uses f(x) of object number 1, d2(x3,x4)f(x) of object number 17, Min2(x3,x4) f(x) of object number 27, Max2(x3,x4) f(x) of object number 37, and ϑ(x3,x4)f(x) of object number 41 for the calculation of ∂2f(x)∂x3∂x4, min2(x4,x5) f(x), max2(x4,x5) f(x), and Δ(x4,x5)f(x), and
1 Reload the TVL-system of Exercise 4.7 as the basis of all further calculations.(M-fold Differential Operations → m-fold Derivative Operations). Calculate all four twofold derivative operations based on the 2-fold differential operations of Exercise 4.7 and verify the results.
Relations for m-fold Derivatives). Calculate all four mfold derivative operations of the functions f(x) (4.9) and g(x), g(x1, x2, x3, x4, x5) = x2(x3 ∨ x4x5) ∨ x1x4(x3 ∨ x5)with regard to (x4, x5), visualize the Karnaugh maps, and verify the 16 relations (4.98), . . . , (4.113) of [18] for
8 Repeat the subtasks 3 . . . 6 for the differential maximum of object number 7.Vectorial Differential Operations). Calculate all vectorial derivative operations based on the partial differential operations of Exercise 4.6 and verify the results. Build the three differential operations of Exercise

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