Questions and Answers of Logic Functions And Equations

4 Calculate ∂f(x)∂x4 using the XBOOLE operation derk, show the Karnaugh map of the result, and compare it with the Karnaugh maps of the given function and the result of subtask 3.
3 Calculate ∂f(x)∂x4 using the XBOOLE operation derv, show the Karnaugh map of the result, and compare it with the Karnaugh map of the given function.
2 Prepare VT x4 as object number 2.
1 Solve the equation f(x1, x2, x3, x4) = 1 for the function (4.7) and store the result as object number 1.
11 Check whether d2(x3,x4)f(x) and ϑ(x3,x4)f(x) are the same function.
10 Check whether dx3f(x) and ϑx3f(x) are the same function.
9 Calculate both ϑx3f(x) as object number 40, and ϑ(x3,x4)f(x) as object number 41, and draw the associated graphs.
8 Execute this PRP and draw the graphs of both m-fold differential maximum operations.
7 Write a PRP that calculates both Maxx3 f(x) as object number 33, and Max2(x3,x4) f(x) as object number 37.
6 Execute this PRP and draw the graphs of both m-fold differential minimum operations.
5 Write a PRP that calculates both Minx3 f(x) as object number 23, and Min2(x3,x4) f(x) as object number 27.
4 Execute this PRP and draw the graphs of the two m-fold differentials.
3 Write a PRP that calculates both dx3f(x) as object number 13, and d2(x3,x4)f(x) as object number 17.
2 Prepare for further calculation the VTs 2: x3, 3: xf3, 4: x4, 5:xf4, and the TVLs for sequential to differential transformation, 6: for x3, and 7: for x4.
Calculate all four m-fold differential operations of the function (4.2) with regard to (x3) and (x3, x4), draw their graphs, compare these graphs with the graphs of the previous Exercise 4.6, and
10 For the last two tasks the function of object number 1 could be used instead of the partial differential expansion F(x1, x2, x3, x4, dx3, dx4).Explain why!
9 Verify the right relation in (4.20) of [18] by means of a DIF-operation between the objects 1 and 7 and store the result as object number 22.
8 Verify the left relation in (4.20) of [18] by means of a DIF-operation between the objects 6 and 1 and store the result as object number 21.
7 Verify relation (4.21) of [18] by means of SYD-operations between the objects 5, 6, and 7 and store the result as object number 20.
6 Create the partial differential maximum Max(x3,x4) f(x) of (4.2) in sequential form as object number 10, apply the PRP for transformation into the differential form, copy the generated object from
5 Create the partial differential minimum Min(x3,x4) f(x) of (4.2) in sequential form as object number 10, apply the PRP for transformation into the differential form, copy the generated object from
4 Create the partial differential d(x3,x4)f(x) of (4.2) in sequential form as object number 10, apply the PRP for transformation into the differential form, copy the generated object from number 11
3 Write a PRP that transforms a differential operation in sequential form, given as object number 10 with regard to the variables (x3, x4), into a differential operation in differential form
2 Create the function f(x1, x2, xf3, xf4) by means of an appropriate PRP and store the orthogonal result as object number 4.
Solve the equation f(x1, x2, x3, x4) = 1 for the function (4.2) and store the result as object number 1.
5 Verify (4.11) of [18] using the total differential minimum of object number 6, the logic function of object number 1, and the total differential maximum of object number 7. Verify the partial order
4 Verify (4.10) of [18] using the total differential maximum of object number 7 and the differential expansion in object number 31. Verify the partial order relation (4.11) of [18] using the results
3 Verify (4.8) of [18] using the total differential minimum of object number 6 and the differential expansion in object number 31. Verify the partial order relation (4.11) of [18] using the results
2 Reload the result of Exercise 4.4 and apply the written PRP in order to generate the total differential expansion of (4.1) based on definition (4.6)of [18]. Visualize the Karnaugh map of the
1 Write a PRP that takes object number 1 as a logic function of three variables (x1, x2, x3) and generates its total differential expansion as object number 31. Verify the partial order relation
8 Compare the Karnaugh maps with the results of Exercise 4.3 and verify relation (4.2) of [18] by means of SYD-operations between the objects 5, 6, and 7 and store the result as object number
7 Create the total differential maximum of (4.1) in sequential form as object number 10, apply the PRP for the transformation into the differential form, copy the generated object from number 11 to
6 Create the total differential minimum of (4.1) in sequential form as object number 10, apply the PRP for transformation into the differential form, copy the generated object from number 11 to
5 Create the total differential of (4.1) in sequential form as object number 10, apply the PRP for the transformation into the differential form, copy the generated object from number 11 to object
4 Write a PRP for the transformation of a differential operation in sequential form given as object number 10 into a differential operation in differential form generated as object number
3 Create the function f(xf1, xf2, xf3) by means of an appropriate CCOoperation.Calculate all three total differential operations of function (4.1) using the transformation method and verify the
2 Prepare VT x1, x2, x3 as object number 2 and VT xf1, xf2, xf3 as object number 3 as the basis for the change of columns.Calculate all three total differential operations of function (4.1) using
1 Solve the equation f(x1, x2, x3) = 1 for function (4.1) and store the result as object number 1.Calculate all three total differential operations of function (4.1) using the transformation method
Calculate all three total differential operations of the function f(x1, x2, x3) = x1(x2 ∨ x3) ⊕ x1(x2 ∼ x3) (4.1)and verify relation (4.2) of [18].6 Verify relation (4.2) of [18] by means of
Calculate all three total differential operations of the function f(x1, x2, x3) = x1(x2 ∨ x3) ⊕ x1(x2 ∼ x3) (4.1)and verify relation (4.2) of [18].5 Calculate the total differential maximum
Calculate all three total differential operations of the function f(x1, x2, x3) = x1(x2 ∨ x3) ⊕ x1(x2 ∼ x3) (4.1)and verify relation (4.2) of [18].4 Calculate the total differential minimum
Calculate all three total differential operations of the function f(x1, x2, x3) = x1(x2 ∨ x3) ⊕ x1(x2 ∼ x3) (4.1)and verify relation (4.2) of [18].3 Calculate the total differential dxf(x) by
Calculate all three total differential operations of the function f(x1, x2, x3) = x1(x2 ∨ x3) ⊕ x1(x2 ∼ x3) (4.1)and verify relation (4.2) of [18].2 Change the expression of f(x) into f(x ⊕
Calculate all three total differential operations of the function f(x1, x2, x3) = x1(x2 ∨ x3) ⊕ x1(x2 ∼ x3) (4.1)and verify relation (4.2) of [18].1 Solve the equation f(x1, x2, x3) = 1 for
5 Execute the PRP e42 gsd.prp and compare whether the result of the inverse transformation (TVL 3) is the same function as the given function(TVL 1) by means of a SYD-operation.Transform the
4 Write a PRP e42 gsd.prp that requires for the inverse transformation the TVL of Gs(a, x, xf) as object number 2 and stores the resulting TVL of Gd(a, x, dx) as object number 3 where intermediate
3 Draw the graph of Gs(a, x, xf) stored as object number 2 and compare it with the graph of Exercise 4.1.Transform the differential representation Gd(a, x, dx) of the graph calculated in Exercise 4.1
1 Write a PRP e42 gds.prp that requires the TVL of Gd(a, x, dx) as object number 1 and stores the resulting TVL of Gs(a, x, xf) as object number 2 where intermediate objects may be stored as objects
Solve the MAXSAT-problem for the following conjunctive form:f = (a ∨ b ∨ c)(b ∨ c)(a ∨ d)(a ∨ d)(b ∨ c)(c ∨ d)(a ∨ c).2 Which disjunctions of the given function must be removed in
Solve the MAXSAT-problem for the following conjunctive form:f = (a ∨ b ∨ c)(b ∨ c)(a ∨ d)(a ∨ d)(b ∨ c)(c ∨ d)(a ∨ c).1 Is the equation f = 1 for the given function satisfiable?
Let be given the function f by a conjunctive form with n = 5 clauses:f = (x1 ∨ x3 ∨ x4)(x2 ∨ x3 ∨ x4)(x2 ∨ x3 ∨ x4)(x1 ∨ x2 ∨ x3)(x1 ∨ x2 ∨ x3).3 Can this problem be solved by
Let be given the function f by a conjunctive form with n = 5 clauses:f = (x1 ∨ x3 ∨ x4)(x2 ∨ x3 ∨ x4)(x2 ∨ x3 ∨ x4)(x1 ∨ x2 ∨ x3)(x1 ∨ x2 ∨ x3).2 Will the solution change when any
Let be given the function f by a conjunctive form with n = 5 clauses:f = (x1 ∨ x3 ∨ x4)(x2 ∨ x3 ∨ x4)(x2 ∨ x3 ∨ x4)(x1 ∨ x2 ∨ x3)(x1 ∨ x2 ∨ x3).1 Represent the same function by k
4 Show that the 5 questions on page 63 of this book can be answered without any special considerations.Let be given the following equation in conjunctive form:(x1 ∨ x3 ∨ x5 ∨ x6)(x3 ∨ x4 ∨
3 Use the SBE operation of XBOOLE to solve this equation and compare this solution with the result of item 1.Let be given the following equation in conjunctive form:(x1 ∨ x3 ∨ x5 ∨ x6)(x3 ∨
2 Use the vectors that are not satisfying the respective single disjunctions and their union and the complement of this set to find the solution.Compare this solution with the result of item 1.Let be
1 Use the orthogonal solution sets for the single disjunctions and the intersection of these sets to find the solution.Let be given the following equation in conjunctive form:(x1 ∨ x3 ∨ x5 ∨
3 Transform the system of equations into a single equation and solve this equation using the operation SBE of the XBOOLE Monitor. Is the solution set equal to the solution set of item 1?Let be given
2 Solve the system of equations using directly the operation SBE of the XBOOLE Monitor. Is the solution set equal to the solution set of item 1?Let be given the system of equations:h1 =x1 ⊕ x2, h2
1 Solve each of these equations separately and calculate the common solution using the partial solution sets.Let be given the system of equations:h1 =x1 ⊕ x2, h2 =x1x2, f1 =h1 ⊕ x3, h3 =h1x3, f2
Exercise 3.8 (Solution with Regard to Variables). Let be given the function f(x1, . . . , xk, xk+1, . . . , xn).1 What can be said about the number of solutions of the equation f(x) = 1 if it is
3 Are there constant functions x3(x1, x2) and x4(x1, x2) which are defined by f(x) = 0?Let be given f(x1, x2, x3x4) = x1x2x3x4 ∨ x1x2x3 ∨ x1x2x3x4 ∨ x1x2x3x4.
2 Are the functions x3(x1, x2) and x4(x1, x2) unique? How the function f(x1, x2, x3, x4) must change in order to make them unique (if they are not unique)?Let be given f(x1, x2, x3x4) = x1x2x3x4 ∨
1 Find functions x3(x1, x2) and x4(x1, x2) which are defined by f(x) = 1.Let be given f(x1, x2, x3x4) = x1x2x3x4 ∨ x1x2x3 ∨ x1x2x3x4 ∨ x1x2x3x4.
Exercise 3.6 (Implication). There are given the functions f(x) = x1 ⊕x2 ⊕ x3 ⊕ x4 ⊕ x5 and g(x) = x1x2x3x4x5 ∨ x1x2x3 ∨ x4x5 ∨ x3x5.1 Check whether f(x) ≤ g(x) holds for the given
Exercise 3.5 (Inequality). Take the functions f(x) and g(x) of Exercise 3.3 and find the solutions of f(x) = g(x). Compare this solution set with the solution set of f(x) = g(x).\
Take the functions f(x) and g(x) of Exercise 3.3 and the solution vector (b1, b2, b3, b4, b5) = (10000) and compare successively the number of solutions of subequations:1 f(x1 = b1) = 1 and g(x1 =
3 Are the sets calculated in the previous two items identical?Let f(x1, x2, x3, x4, x5)=((((x1 ↓ x2) ↓ x3) ↓ x4) ↓ x5), g(x1, x2, x3, x4, x5)=((((x1|x2)|x3)|x4)|x5).
2 Solve the equation f(x) = g(x) directly using the procedure implemented in the XBOOLE Monitor.Let f(x1, x2, x3, x4, x5)=((((x1 ↓ x2) ↓ x3) ↓ x4) ↓ x5), g(x1, x2, x3, x4,
1 Find all solution vectors of the equation f(x) = g(x) by means of set operations.Let f(x1, x2, x3, x4, x5)=((((x1 ↓ x2) ↓ x3) ↓ x4) ↓ x5), g(x1, x2, x3, x4, x5)=((((x1|x2)|x3)|x4)|x5).
Exercise 3.2 (Partial Solutions). For the given functions f(x) and g(x) of Exercise 3.1 consider the sets f0, g0, f1 und g1 and build (by set operations)(f0 ∩ g0) ∪ (f1 ∩ g1). What can be said
5 Compare this set with the solution set of f(x1, x2, x3, x4, x5) ∼ g(x1, x2, x3, x4, x5) = 1. Let f(x1, x2, x3, x4, x5) = x2x3x4 ∨ x2x3x5 ∨x1x3x5, g(x1, x2, x3, x4, x5) = (x2 ⊕ x3).
4 Compare this set with the solution set of f(x1, x2, x3, x4, x5) ∼ g(x1, x2, x3, x4, x5) = 0. Let f(x1, x2, x3, x4, x5) = x2x3x4 ∨ x2x3x5 ∨x1x3x5, g(x1, x2, x3, x4, x5) = (x2 ⊕ x3).
Compare this set with the solution set of f(x1, x2, x3, x4, x5)⊕g(x1, x2, x3, x4, x5) = 1. Let f(x1, x2, x3, x4, x5) = x2x3x4 ∨ x2x3x5 ∨x1x3x5, g(x1, x2, x3, x4, x5) = (x2 ⊕ x3).
2 Compare this set with the solution set of f(x1, x2, x3, x4, x5)⊕g(x1, x2, x3, x4, x5) = 0. Let f(x1, x2, x3, x4, x5) = x2x3x4 ∨ x2x3x5 ∨x1x3x5, g(x1, x2, x3, x4, x5) = (x2 ⊕ x3).
Let f(x1, x2, x3, x4, x5) = x2x3x4 ∨ x2x3x5 ∨x1x3x5, g(x1, x2, x3, x4, x5) = (x2 ⊕ x3).1 Find all solution vectors of the equation f(x) = g(x).
A graph G is given by means of the adjacency matrix M:This adjacency matrix describes a graph with 10 nodes. The element M1,2 = 1 describes an edge from node 1 to node 2 etc. By the way, this graph
Let be given the following graph by its adjacency matrix:1 Draw a sketch of this graph.2 Design a Boolean model of the problem of extended Hamiltonian graphs and fit it into a PRP.3 Execute the PRP
Exercise 7.1 (Structural Model – System of Logic Equations). Create a system of logic equations as structural model of the combinatorial circuit shown as schematic diagram in Fig. 7.1. Prepare this
Exercise 7.2 (Structural Model – Set of Local Lists of Phases). Create a set of local lists of phases as structural model of the combinatorial circuit shown as schematic diagram in Fig. 7.1.
Exercise 7.3 (Structural Model – TVL in a Certain Form). Assume the circuit structure in Fig. 7.1 may by transformed into a two level AND-OR structure such that the conjunctions of the disjunctive
Exercise 7.4 (Behavioral Model – Explicit Equation System – System Function of a Completely Specified Circuit). The behavior of a circuit is given by the equation system (7.1),Calculate a
Exercise 7.5 (Behavioral Model – System of Function TVLs of a Completely Specified Circuit). Separate the system function of Exercise 7.4 into a system of function TVLs using formula (7.56) of
1 Prepare a Boolean space and attach the input and output variables in a convenient order. Verify that the system function F(x, y) (7.2) describes an incompletely specified function.F(x, y)=((x1x2(x3
2 Solve (7.2). Verify that the system function F(x, y) (7.2) describes an incompletely specified function.F(x, y)=((x1x2(x3 ⊕ x4)y ∨ x3x4(x5 ⊕ x6)y ∨ x1x7y∨ x1x2x6x7) ⊕ x6) ∨ (x3 ⊕
3 Prepare solutions of the simple equations y = 1. Verify that the system function F(x, y) (7.2) describes an incompletely specified function.F(x, y)=((x1x2(x3 ⊕ x4)y ∨ x3x4(x5 ⊕ x6)y ∨
4 Calculate the mark function fϕ(x) of the don’t-care set based on formula(7.24) of [18]. Verify that the system function F(x, y) (7.2) describes an incompletely specified function.F(x,
5 Calculate the mark function fq(x) of the ON-set based on formula (7.25)of [18]. Verify that the system function F(x, y) (7.2) describes an incompletely specified function.F(x, y)=((x1x2(x3 ⊕ x4)y
6 Calculate the mark function fr(x) of the OFF-set based on the formula(7.25) of [18]. Verify that the system function F(x, y) (7.2) describes an incompletely specified function.F(x, y)=((x1x2(x3 ⊕
7 Verify whether the three mark functions are pairwisely disjoint and cover the Boolean space completely. Verify that the system function F(x, y) (7.2) describes an incompletely specified
8 How many functions includes the characteristic function set FC(x) which is specified by the calculated mark functions? Verify that the system function F(x, y) (7.2) describes an incompletely
9 Calculate the system function using the mark functions fq(x) and fr(x)based on formula (7.21) of [18] and verify the result. Verify that the system function F(x, y) (7.2) describes an incompletely
10 Calculate the system function using the mark functions fϕ(x) and fr(x)based on formula (7.22) of [18] and verify the result. Verify that the system function F(x, y) (7.2) describes an
11 Calculate the system function using the mark functions fq(x) and fϕ(x)based on formula (7.23) of [18] and verify the result. Verify that the system function F(x, y) (7.2) describes an
Exercise 7.7 (Behavior of a Combinatorial Circuit Based on a System of Logic Equations). Calculate the behavior of the circuit given in Fig. 7.1.Use the system of logic equations prepared in Exercise
Exercise 7.8 (Behavior of a Combinatorial Circuit Based on a Set of Local Lists of Phases). Calculate the behavior of the circuit given in Fig. 7.1.Use the set of local lists of phases prepared in
Exercise 7.9 (Input-Output Behavior of a Combinatorial Circuit). Calculate the input-output behavior of the circuit given in Fig. 7.1. Minimize the result to an orthogonal TVL. Practical tasks:1
Exercise 7.10 (Simulation Based on a Global List of Phases). Execute several simulations of a modified circuit created from the circuit structure in Fig. 7.1. The circuit is extended such that
Exercise 7.11 (Don’t-Care Function Defined by the Outputs of a Global List of Phases). Calculate both the output pattern of the circuit modified in Exercise 7.10 and the don’t-care function
Exercise 7.12 (Independence of a Function Regarding Variables). Analyze whether the functions y1(x) and y2(x) calculated in Task 3 of Exercise 7.5 really depend on the variables (x1, x2, x3, x4, x5,

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