Questions and Answers of Logic Functions And Equations

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
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
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
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
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
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
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
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
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
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)
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,
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
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
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
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
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);
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
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) |
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)
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
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
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
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
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
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
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
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
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
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
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 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 ∨
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
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 ∨
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) =
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
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
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
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
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,
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
7 Repeat the subtasks 3 . . . 6 for the differential minimum of object number 6.Vectorial Differential Operations). Calculate all vectorial derivative operations based on the partial differential
6 Execute the PRP of the previous subtask.Vectorial Differential Operations). Calculate all vectorial derivative operations based on the partial differential operations of Exercise 4.6 and verify the
5 Write a PRP that calculates partial differentials of the function in object number 1 using their vectorial derivatives, calculated in the previous PRP, and verify the result.Vectorial Differential
4 Execute the PRP of the previous subtask.Vectorial Differential Operations). Calculate all vectorial derivative operations based on the partial differential operations of Exercise 4.6 and verify the
3 Write a PRP that calculates all vectorial derivatives using the partial differential, given in object number 5, and verify these results using vectorial derivatives of the function in object number
2 Prepare TVLs for conjunctions dx3 dx4, dx3 dx4, dx3 dx4, and dx3 dx4, as objects number 30, . . . , 33, and VTs of x3, x4, and x3, x4 as objects number 34, 35, and 36.Vectorial Differential
1. Reload the TVL-system of Exercise 4.6 as the basis of all further calculations.Vectorial Differential Operations). Calculate all vectorial derivative operations based on the partial differential
7 Verify (4.43) of [18]. Calculate all three vectorial derivatives of the function (4.7) with regard to (x3, x4) using the XBOOLE operations mentioned above, visualize the Karnaugh maps, observe that
6 Verify (4.52) of [18]. Calculate all three vectorial derivatives of the function (4.7) with regard to (x3, x4) using the XBOOLE operations mentioned above, visualize the Karnaugh maps, observe that
5 Calculate max(x3,x4) f(x) as object number 5, and show the Karnaugh map of the result. Calculate all three vectorial derivatives of the function (4.7) with regard to (x3, x4) using the XBOOLE
4 Calculate min(x3,x4) f(x) as object number 4, and show the Karnaugh map of the result. Calculate all three vectorial derivatives of the function (4.7) with regard to (x3, x4) using the XBOOLE
3 Calculate ∂f(x)∂(x3,x4) as object number 3, and show the Karnaugh map of the result. Calculate all three vectorial derivatives of the function (4.7) with regard to (x3, x4) using the XBOOLE
2 Prepare VT x3, x4 as object number 2. Calculate all three vectorial derivatives of the function (4.7) with regard to (x3, x4) using the XBOOLE operations mentioned above, visualize the Karnaugh
Calculate all three vectorial derivatives of the function (4.7) with regard to (x3, x4) using the XBOOLE operations mentioned above, visualize the Karnaugh maps, observe that the vectorial
11 Verify (4.48) of [18] and emphasize the understanding of this relation by comparing Karnaugh map 4, 3, and 5.
10 Verify (4.47) of [18] and emphasize the understanding of this relation by comparing Karnaugh map 5, 3, and 4.
9 Verify (4.46) of [18] and emphasize the understanding of this relation by comparing Karnaugh map 5, 4, and 3.
8 Verify (4.45) of [18] and emphasize the understanding of this relation by comparing Karnaugh map 3 and 4.
7 Verify (4.44) of [18] and emphasize the understanding of this relation by comparing Karnaugh map 3 and 5.
6 Verify (4.43) of [18] and emphasize the understanding of this relation by comparing Karnaugh map 4 and 5.
5 Calculate maxx4 f(x) as object number 5, and show the Karnaugh map of the result.
4 Calculate minx4 f(x) as object number 4, and show the Karnaugh map of the result.
3 Calculate ∂f(x)∂x4 as object number 3, and show the Karnaugh map of the result.
1 Solve the equation f(x1, x2, x3, x4) = 1 for function (4.7) and store the result as object number 1.
Calculate all three simple derivatives of the function (4.7) with regard to x4 using m-fold XBOOLE derivative operations for m = 1, visualize the Karnaugh maps, and verify the six relations (4.43), .
10 Verify (4.40) of [18].
9 Verify (4.36) of [18].
8 Calculate maxx4f(x) using the XBOOLE operation maxk, show the Karnaugh map of the result, and compare it with the Karnaugh maps of the given function and the result of subtask 7.
7 Calculate maxx4f(x) using the XBOOLE operation maxv, show the Karnaugh map of the result, and compare it with the Karnaugh map of the given function.
6 Calculate minx4f(x) using the XBOOLE operation mink, show the Karnaugh map of the result, and compare it with the Karnaugh maps of the given function and the result of subtask 5.
5 Calculate minx4f(x) using the XBOOLE operation minv, show the Karnaugh map of the result, and compare it with the Karnaugh map of the given function.

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