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logic functions and equations

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

Show that the following rules are always valid. Translate these rules into “natural language”.8 (z → x) → ((z → y) → (z → (x ∧ y)));9 ((x ∧ y) ∨ z) → ((x ∨ z) ∧ (y ∨ z)), ((x ∨ z) ∧ (y ∨ z)) → ((x ∧ y) ∨ z);10 ((x ∨ y) ∧ z) → ((x ∧ z) ∨ (y ∧ z)),
Show that the following rules are always valid. Translate these rules into “natural language”.1 (x → x);2 x → (y → x);3 (x → y) → ((y → z) → (x → z));4 (x → (y → z)) → ((x → y) → (x → z));5 x → (x ∨ y), y → (x ∨ y);6 (x → z) → ((y → z) → ((x ∨ y)
Verify the following rules and explain these rules in natural language:5 (x ∨ y) ∧ x → y disjunctive syllogism;6 (y → z) → ((x ∧ y) → z) strengthening the antecedent;7 (x → y) → (x → (y ∨ z)) weakening the consequent;8 (x → y) ∧ (x → y) → y;9 (x → y) ∧ (x → y)
Verify the following rules and explain these rules in natural language:1 x ∧ (x → y) → y modus ponens;2 y ∧ (x → y) → x modus tollens;3 (x → y) → (y → x) contrapositive;4 (x → y) ∧ (y → z) → (x → z) hypothetical syllogism;
Exercise 6.5 (Rules). Explain the following rules in natural language:1 x ∨ x is a tautology – Law of the excluded middle;2 x ∨ (y ∨ z) → (x ∨ y) ∨ z – Associative rule;3 x → (x ∨ y) – Expansion rule;4 (x ∨ x) → x – Contraction rule;5 (x ∨ y) ∧ (x ∨ z) → (y ∨
Show the paradoxes of the implication by confirming that the following rules are tautologies.3 (x → y) ∨ (y → x). Of any two propositions, one implies the other.
Show the paradoxes of the implication by confirming that the following rules are tautologies.2 x → (x → y). A false proposition x implies any proposition y.
Show the paradoxes of the implication by confirming that the following rules are tautologies.1 x → (y → x). A true proposition x is implied by any proposition y.
2 Translate this expression into a set of assumptions and a set of rules.Express the meaning of this form in natural language.The argument (x1 → x2) ∧(x3 → x4) ∧ (x1 ∨ x3) → (x2 ∨ x4) is given.
1 Show that this argument form is valid.The argument (x1 → x2) ∧(x3 → x4) ∧ (x1 ∨ x3) → (x2 ∨ x4) is given.
Show that the following rules are always valid! Translate these rules into “natural language”!1 (x → y) ∼ (x ∨ y);2 (x → y) ∼ (x ∧ y);3 (x → y) ∼ (x ∨ (x ∧ y))
2 Solve this system of equations by using and combining the solution sets of the three single equations.Let be given the following system of equations:x1 ⊕ x3 ⊕ x6 = 0, x2 ⊕ x4 ⊕ x6 = 0, x1 ⊕ x2 ⊕ x4 ⊕ x5 = 1.
1 Combine these three equations into one equation and use the XBOOLE Monitor for a solution in one step, after the possible simplification of the equation.Let be given the following system of equations:x1 ⊕ x3 ⊕ x6 = 0, x2 ⊕ x4 ⊕ x6 = 0, x1 ⊕ x2 ⊕ x4 ⊕ x5 = 1.
3 Combine considerations of the right sides and the values of variables together with required (partial) solution sets Let be given the following system of equations:x1 ∨ x3 ∨ x6 = 0, x2 ∨ x4 ∨ x6 = 0, x1 ∨ x2 ∨ x4 ∨ x5 = 1.
2 Solve this system of equations by using and combining the solution sets of the three single equations. Let be given the following system of equations:x1 ∨ x3 ∨ x6 = 0, x2 ∨ x4 ∨ x6 = 0, x1 ∨ x2 ∨ x4 ∨ x5 = 1.
1 Combine these three equations into one equation and use the XBOOLE Monitor for a solution in one step. Let be given the following system of equations:x1 ∨ x3 ∨ x6 = 0, x2 ∨ x4 ∨ x6 = 0, x1 ∨ x2 ∨ x4 ∨ x5 = 1.
3 Explain the use of the symmetric difference and the complement of the symmetric difference. Let be given the two equations x1 ⊕ x1x2 ⊕ x1x2x3 ⊕ x1x2x3x4 ⊕ x1x2x3x4x5 = 0 and x1 ∼ x1x2 ∼ x1x2x3 ∼ x1x2x3x4 ∼ x1x2x3x4x5 = 0
2 Verify your result using the XBOOLE Monitor. Let be given the two equations x1 ⊕ x1x2 ⊕ x1x2x3 ⊕ x1x2x3x4 ⊕ x1x2x3x4x5 = 0 and x1 ∼ x1x2 ∼ x1x2x3 ∼ x1x2x3x4 ∼ x1x2x3x4x5 = 0
1 Compare the solution sets of these two equations by means of transforming the given expressions. Let be given the two equations x1 ⊕ x1x2 ⊕ x1x2x3 ⊕ x1x2x3x4 ⊕ x1x2x3x4x5 = 0 and x1 ∼ x1x2 ∼ x1x2x3 ∼ x1x2x3x4 ∼ x1x2x3x4x5 = 0
2 What is the required set operation for the equations f(x) ∧ g(x) = 0 and f(x) ∨ g(x) = 1? Let be given partial solution sets of f(x) = 1, g(x) = 1, f(x) = 0, and g(x) = 0, respectively. Consider the two equations f(x) ∧ g(x) = 1 and f(x) ∨ g(x) = 0.
1 Why in both cases the intersection of partial solution sets with the same right side has to be used? Let be given partial solution sets of f(x) = 1, g(x) = 1, f(x) = 0, and g(x) = 0, respectively. Consider the two equations f(x) ∧ g(x) = 1 and f(x) ∨ g(x) = 0.
3 Verify the results by comparison of the previous two subtasks Let be given f(x) = (((x1 → x2) ↓ x3)|x4)⊕ x5.
2 Solve these equations by operations with sets. Let be given f(x) = (((x1 → x2) ↓ x3)|x4)⊕ x5.
1 Solve the equations f(x) = 0 and f(x) = 1 by using the XBOOLE Monitor in one step.Hint: transfer | (NAND) and ↓ (NOR) into formulas that can be understood by the XBOOLE Monitor.Let be given f(x) = (((x1 → x2) ↓ x3)|x4)⊕ x5.
Exercise 2.45. Let be given the function f(x1, x2, x3, x4) = x1x2∨x2x3∨x3x4 ∨ x4x1. However, it is not defined for (1110) and (1111).1 Find the set of four functions that meet this partial specification.2 Find the function ϕ(x1, x2, x3, x4) for this situation.3 Represent the four possible
Exercise 2.44. Find the function ϕ(x1, x2, x3, x4) for the following definition ranges:1 The function f(x1, x2, x3, x4) is defined only for vectors with one or three values 1.2 The function f(x1, x2, x3, x4) is not defined for all vectors with two values 1.3 The function f(x1, x2, x3, x4) is not
Exercise 2.43. Let be given the function(s) (10−−010 − 11000101).1 Find the set of functions that can be derived from this partial definition.2 Find for every function the antivalence normal form and compare the different representations.3 Find for every function the conjunctive normal form
Exercise 2.41. Represent every function of this exercise using all the complete systems given in Exercise 2.39.1 f1 = x1 ⊕ x2 ⊕ x3 ⊕ x4 ⊕ x5;2 f2 = (x1 ∨ x2 ∨ x3)(x1 ∨ x2 ∨ x3) ⊕ x4 ⊕ x5;3 f3 = (x1 ∨ x2 ∨ x3)(x1 ∨ x2 ∨ x3)(x4 ⊕ x5);4 f4 = (x1 ⊕ x2 ⊕ x3)(x4 ⊕ x5
Exercise 2.40. Express every function that appears in Exercise 2.39 by all the other complete systems of functions of this exercise
Exercise 2.39. Show that the following systems of functions are complete systems:1 {x ↓ y};2 {xy ⊕ z, (x ∼ y) ⊕ z};3 {x → y, x ⊕ y ⊕ z};4 {x → y, (1100001100111100)};5 {0, xy ∨ xz ∨ yz, 1 ⊕ x ⊕ y ⊕ z};6 {(1011), (1111110011000000)}.
Exercise 2.38 (Essential Variables). Find the essential or non-essential variables of the following functions and find formulas without non-essential variables:1 f(x1, x2, x3) = (x1 → (x1 ∨ x2)) → x3;2 f(x1, x2) = ((x1 ∨ x2) → x2);3 f(x1, x2, x3, x4) = (x1 ∨ x2 ∨ x2x3 ∨ x1x2x3)x4;4
Exercise 2.37 (Minimized Disjunctive Normal Form). What can be said about the minimization of the following functions:1 f1 = x1 ⊕ x2 ⊕ x3 ⊕ x4 ⊕ x5;2 f2 = (x1 ∨ x2 ∨ x3)(x1 ∨ x2 ∨ x3) ⊕ x4 ⊕ x5;3 f3 = (x1 ∨ x2 ∨ x3)(x1 ∨ x2 ∨ x3)(x4 ⊕ x5);4 f4 = (x1 ⊕ x2 ⊕ x3)(x4
Exercise 2.36 (Prime Implicants). Find the prime implicants of the following functions:1 f(x, y, z) = (00101111);2 f(x, y, z) = (01111110);3 f(x1, x2, x3, x4) = (1010111001011110).
Exercise 2.35 (Functional Constraints). Find all functions satisfying the following conditions:1 f(1, 0, 0, 0) = 1, f(0, 1, 1, 1) = 0;2 f(1, 0, 0, 0) = 1, f linear in one or more variables;3 f(0, 1, 0, 0) = f(1, 0, 1, 1), f symmetric (consider all possibilities);4 f(1, 0, 0, 1) = 0, f self-dual
Exercise 2.34 (Monotone Functions). Let be given f(x1, x2, x3, x4) such that f(0, 1, 1, 0) = 1, f(1, 1, 0, 0) = 1, f(1, 0, 1, 0) = 0, f(0, 0, 1, 1) = 1, f(0, 1, 0, 1) = 0.1 Can this definition be used to build monotone functions with these values?2 How many different monotone functions with these
Exercise 2.33 (Monotone Functions). For each monotone function, we have f(x) = xif(xi = 1) ∨ f(xi = 0), and f(x) = (xi ∨ f(xi = 0)) ∧ f(xi = 1).Prove these identities.
Exercise 2.32 (Monotone Functions). For which value of n are the following functions monotone?1 f1(x1, . . . , xn) = x1x2 ∨ x1x3 ∨ · · · xn−1xn (the disjunctions of all conjunctions consisting of two non-negated variables);2 f2(x1, . . . , xn) = x1x2 . . . xn−1xn → (x1 ⊕ x2 ⊕· ·
Exercise 2.31 (Monotone Functions). Which functions of the given set are monotone? Check the increasing as well as the decreasing possibility.1 f1 = (x → (x → y)) → (y → z);2 f2 = x1x2 ⊕ x1x3 ⊕ x1x4 ⊕ x2x3 ⊕ x2x4 ⊕ x3x4;3 f3 = (0000000010111111);4 f4 = (0001010101010111).
Exercise 2.30 (Symmetric Functions). 1 Find the number of functions f(x1, . . . , xn) which are symmetric in (x1, x2), n ≥ 2.2 Find all functions that do not change their values for any permutation of the variables.
Exercise 2.29 (Symmetric Functions). 1 Find functions f(x1, x2, x3)symmetric in (x1, x2).2 Find functions f(x1, x2, x3) symmetric in (x2, x3).3 What can be said about the intersection of these two sets?
Exercise 2.28 (Dual and Self-Dual Functions). Let be given the following functions:f1 = (11100111);f2 = (01110001);f3 = (11001101);f4 = x1x2 ∨ x2(x3 ⊕ x4).Find for each function its dual function! Is one of these functions a self-dual function?
Find all linearly degenerated functions of three and four variables using the equivalence. Explain the construction! What is the common property of these functions?
Find all linearly degenerated functions of three and four variables using the antivalence. Explain the construction! What is the common property of these functions?
Find all disjunctively degenerated functions of three and four variables.Explain the construction! What is the common property of these functions?
Find all conjunctively degenerated functions of three and four variables. Explain the construction!What is the common property of these functions?
2 Which functions f(x1, x2) do not change the value when x1 and x2 are exchanging their positions?
1 For f = ((x1 → x2x3) ⊕ x2)x2, find subfunctions f1(x2, x3) = f(x1 = 1, x2, x3), f2(x1, x3) = f(x1, x2 =1, x3), f3(x3) = f(x1 = 1, x2 = 0, x3) using the intersection and the insertion of constants.
Exercise 2.25 (Shegalkin Polynomials). 1 Transform the antivalence forms of the previous problem into Shegalkin polynomials.2 Find the Shegalkin polynomial for f = x1 ∨ x2 ∨ x3.
Exercise 2.23 (Special Normal Forms). How many disjunctions (conjunctions)will be used for the conjunctive (disjunctive) normal forms of the following functions:1 f = x1 ⊕ x2 ⊕· · ·⊕xn;2 g = (x1 ∨ x2 ∨ · · · ∨ xn)(x1 ∨ x2 ∨ · · · ∨ xn);3 h = (x1 ∨ x2 ∨ x3)(x1 ∨ x2
Exercise 2.22 (Function Vectors). 1 Find the disjunctive and the conjunctive normal forms for the two given function vectors.2 Find shorter disjunctive and conjunctive forms using the Karnaugh map.3 Use the items OBB Orthogonal Block Building and OBBC Orthogonal Block Building and Change of the
(NAND and NOR). 3 What can be said about f = x1|x2|x3|x4 and g = x1 ↓ x2 ↓ x3 ↓ x4?
3 Discuss the result received for f1 = x1 → x2 → x3 → x4 → x5 → x6 →x7 → x8 → x9 → x10.
2 Which function is the XBOOLE Monitor using for x → y → z.
1 Find the TVLs for the functions f =(x → y) → z and g = x → (y → z).
Show the TVL and the Karnaugh map of the following function:4 f4 = x1 ∼ x2 ∼ x3 ∼ x4 ∼ x5 ∼ x6 ∼ x7 ∼ x8 ∼ x9 ∼ x10.
Show the TVL and the Karnaugh map of the following function:3 f3 = x1 ⊕ x2 ⊕ x3 ⊕ x4 ⊕ x5 ⊕ x6 ⊕ x7 ⊕ x8 ⊕ x9 ⊕ x10;
Show the TVL and the Karnaugh map of the following function:2 f2 = x1 ∨ x2 ∨ x3 ∨ x4 ∨ x5 ∨ x6 ∨ x7 ∨ x8 ∨ x9 ∨ x10;
Show the TVL and the Karnaugh map of the following function:1 f1 = x1x2x3x4x5x6x7x8x9x10;
Let be given f(a,b) = a ∨ b and g(x3, x4) = x3 ∼ x4.3 Find all vectors (x1, x2, x3, x4) with h(x1, x2, x3, x4) = f(x1, x2) ∧g(x3, x4).
Let be given f(a,b) = a ∨ b and g(x3, x4) = x3 ∼ x4.2 Find all vectors (x1, x2, x3, x4) with h(x1, x2, x3, x4) = f(x1, x2) ∨g(x3, x4).
Let be given f(a,b) = a ∨ b and g(x3, x4) = x3 ∼ x4.1 Find all vectors (x2, x3, x4) with h(x2, x3, x4) = f(g(x3, x4), x2).
Find the Karnaugh map, a TVL and a disjunctive form for the following functions!1 The function f(x, y, z) has the value 1 either for x = 1 or if y = z and the value of x is less than the value of z, otherwise the value of the function is equal to 0.2 f(x1, x2, x3, x4) = 0 for such vectors
3 How many functions of n variables exist with less than k values 1, k ≥ 1?
2 Two binary vectors x = (x1, . . . , xn) and y = (y1, . . . , yn) are called neighbored if h(x, y) = 1. How many functions exist with f(x) = f(y)for neighbored vectors x and y?
Which functions are represented by solving the equations (x ∧ y) = 0 and (x ∨ y) = 0, resp. Use the Karnaugh map to define an appropriate expression for these functions
What is the relation between spheres with center c and spheres with center c? Base your considerations on the analogous relation for shells.
Let n = 4, c = (0000). Show that K0 = S0, . . . ,K4 = S0 ∪ · · · ∪ S4.Generalize this relation to any value of n.
Let n = 4, c = (0000). Show that K0 = ∅, K1 = S0, . . . , K5 =S0 ∪ · · · ∪ S4. Generalize this relation to any value of n.
Confirm that Si (x,c) = Sn−i(x, c).
Now use c = (1111) and find Si(x, c), i = 0, . . . , 4 with regard to this new center.
Use c = (0000) and find Si(x, c)for i = 0, . . . , 4 with regard to this center.
Activate in the menu labeled by ‘?’ the item Help Topics. Select in the dialog window that appears now in the tab Content the item Graphical User Interface and there the item Menus. Study the meaning of the menu items.In the initial state of the XBOOLE Monitor all items in the menus Derivative,
Define two Boolean spaces using the menu.The number of variables in the first space must be 32 and in the second space 512, respectively. Visualize the results in the tab labeled by Spaces/Objects. Explain the values Type and Variables.

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