* b) Show (p = q) + (~p V q)
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- 5 A. Let * be a new logical connective such that p * q does not hold if and only if p and q are either both false or both true. (i) Write down the truth table for p * q (ii) Write down the truth table for (p * p) * (q * q). (iii) Does the table in (b) coincide with a table in *Truth table available on the given image* (By "Coincide" we mean that the respective columns of T and F values are the same) B. Construct a formula in CNF based on each of the following truth tables: *Truth table available on the given image* C. Compute CNF (NNF ( IMPL_FREE ¬(p → ( ¬ (q ∧ ( ¬p → q))))))Consider the following language assuming ∑ = {a, b} The total number of a's are even including null The a's can be either consecutive or non-consecutive The number of b's are not restricted, i.e. any number of b's Build an FA using minimum number of states. Write a regular expression. Justify your answers with examples (abbabaa, aaaa, abababab bbb are acceptd; bababbba and bbbaaa, aaa are NOT accepted).Q. There is an island that has two kinds of inhabitants, knights, who always tellthe truth, and their opposites, knaves, who always lie. It is assumed that every inhabitant of theisland is either a knight or a knave. Below there are 3 inhabitants, who are denoted by A, B andC.A. What are A, B and C if A says “If B is a knave then C is a knave”, and B says “If Cis a knight then A is a knave”? Briefly explain your reasoning.B. What are A,B and C if A says “B is a knight and C is a knight”, and B says “A is aknight if and only if C is a knave”? Briefly explain your reasoning.
- A self-avoiding walk in a lattice is a path from one point to another that does not visit the same point twice. Self-avoiding walks have applications in physics, chemistry, and mathematics. They can be used to model chainlike entities such as solvents and polymers. Write a Turtle program that displays a random path that starts from the center and ends ata point on the boundary, as shown in Figure 11.11a, or ends at a dead-end point (i.e., surrounded by four points that have already been visited), as shown in Figure 11.11b. Assume the size of the lattice is 16 * 16.Determine whether the following is true or false. Please cite the brief explanation so that I can know why is it true or false. a) The sentence “If x > 0, then compute √π”, is as tatement. b) The compound statement p ∧(q ∨¬p )∧¬q is ac ontradiction. c) ¬( p ∨ q) ≡ ¬p ∧¬q d) 1 ∈ {{1},{2},{3}} e) Set B = {x | x ∈ ℝ and x4 =16} is finite. f) [2,4] U (3,5) is a disjoint union g) A\B = A ∩ Bc h) A X B = B X A I) {{Ø}} = 0 j) The contrapositive of p -> (p V ¬p) = (¬p∧ q) -> ¬p4. Let N(x) be the statement “x has visited North Dakota,” where the domain consists of the students in your school. Express each of these quantifications in English. a) ∃x N(x) b) ∀x N(x) c) ˺∃x N(x) d) ∃x ˺N(x) e) ˺∀x N(x) f ) ∀x ˺N(x)
- Q 3.I] Write a Java program to read the length and breadth using console and find the area of the rectangle. If the area is greater than 5oo then find the Perimeter of the rectangle. Area of rectangle=length∗breadth Perimeter of the rectangle=2∗(length breadth)Part I. Let p, q, and r be the following simple statements: p: Sydney is the capital of Australia. q: Thirteen is a prime number. r: The Trinity University of Asia is in Quezon City. Express each of the following propositions as an English sentence and determine its truth value. Part II. Use the truth table to determine whether the following pairs of statement are logically equivalent or not . 1. ~(p↑q)and~p↑-q 2. p↓(q↑r)and(p↑q)↓(p↑r)Propositions and quantifiers Let x, y ∈ {2, 3, 4}. For the following propositions provide aproof or counterexample to the statements below:•P0(x, y): x < y•P1(x, y): both x and y are even•P2(x, y): x ×y > 7•P3(x, y): x is even and x + y is a multiple of x.•P4(x, y): x ̸= y (a) for all propositions P in P0, . . . , P4,[∃x ∀y P ] →[∀y ∃x P ] (b) for all propositions P in P0, . . . , P4,[∀x ∃y P ] →[∃y ∀x P ]
- Not pseudocode Suppose the economies of the world use a set of currencies C1, . . . , Cn; think of these as dollars, pounds, Bitcoin, etc. Your bank allows you to trade each currency Ci for any other currency Cj, and finds some way to charge you for this service. Suppose that for each ordered pair of currencies (Ci, Cj ), the bank charges a flat fee of fij > 0 dollars to exchange Ci for Cj (regardless of the quantity of currency being exchanged). Describe an algorithm which, given a starting currency Cs, a target currency Ct, and a list of fees fij for all i, j ∈ {1, . . . , n}, computes the cheapest way (that is, incurring the least in fees) to exchange all of our currency in Cs into currency Ct. Also, justify the its runtime. [We are expecting a description of the algorithm, as well as a brief justification of its runtime.]3. It can be said that statement P is a necessary condition for statement Q if Q ⇒ P is true. In other words, in order for Q to be true, it is necessary that P must be true. Let x be an integer. Give a necessary but not sufficient condition on x for x+5 to be a prime number. Justify your answer. Note: You should give a condition related to x, but not picking one specific number for x.f : {1, 2, 3} ® {a, b, c, d} f(1) = c f(2) = b f(3) = a g : {a, b, c, d} ® {x, y, z} g(a) = y g(b) = x g(c) = x g(d) = z Find the composition gof