a) Prove the assignment segment given below to its pre-condition and post-condition using Hoare triple method. (t >- 5) void main () int f,g,2 2- 2t- 8: g- 2 21 f- 2g - 5: If >- 3) b) Identify the weakest pre-condition for the following decision that can guarantee the post- condition to be true. (?) integer m, n if (m > n) y-8/y+ 1 else y - 3 y y 29) c) 1. Find the loop invariant for the program code below. void main () int y - 1, x - 0: while (x < 4) y- y a X - X 1i
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- Consider the following statement. If n is divisible by 6, then n is divisible by 2 and n is divisible by 3. In (a)–(d) below, select the negation, contrapositive, converse, and inverse for the statement. (Assume that all variables represent fixed quantities or entities, as appropriate.) (a) Negation n is divisible by 6 and either n is not divisible by 2 or n is not divisible by 3.If n is not divisible by 2 or n is not divisible by 3, then n is not divisible by 6. If n is divisible by 2 and n is divisible by 3, then n is divisible by 6.If n is not divisible by 6, then n is not divisible by 2 or n is not divisible by 3.If n is not divisible by 6, then n is divisible by 2 and n is divisible by 3. (b) Contrapositive n is divisible by 6 and either n is not divisible by 2 or n is not divisible by 3.If n is not divisible by 2 or n is not divisible by 3, then n is not divisible by 6. If n is divisible by 2 and n is divisible by 3, then n is divisible by 6.If n is not divisible by 6, then…Exercise 1.4.3: Proving two logical expressions are not logically equivalent. About Prove that the following pairs of expressions are not logically equivalent. (a) p → q and q → p (b) ¬p → q and ¬p ∨ q (c) (p → q) ∧ (r → q) and (p ∧ r) → q (d) p ∧ (p → q) and p ∨ qComplete the following truth table by filling in the blanks with T or F as appropriate. p q r ?→? ?→? [?→?]∨[?→?] ?∨? ?→[?∨?] T T T T T F T F T T F F F T T F T F F F T F F F "[?→?]∨[?→?][p→q]∨[p→r]" and "?→[?∨?]p→[q∨r]" are A. logically equivalent B. not logically equivalent C. not logically comparable
- Answer all 3 must else ?Explain the four criteria for proving the correctness of a logical pretestloop construct of the form while B do S end.Let ? = {?| ? ?? ?? ??????? ??? ?2 < 16}. Identify each of the following as true or false. Explainyour answer in each case.1.1 { } ⊆ ? 1.2 {0, 1, 2, 3} ⊆ ? 1.3 {−3, −2, −1} ⊆ ? 1.4 ? ⊆ {−3, −2, −1, 0, 1, 2, 3} (1.5 { |?? ?? ??????? ??? |?| < 4} ⊆ C
- Please don't provide handwritten solution ........ For each of the following functions, indicate the class Θ(g(n)) the function belongs to. (Use the simplest g(n) possible in your answers.) Prove your assertions. a. √10n4 + 7n3 + 3n2 +n+10 b. 2n+1 +3n−1"Equatable and Comparable" in the Swift Programming: The Big Nerd Ranch Guide (2nd Ed.) e-book: Point’s current conformance to Comparable yields some confusing results. let c = Point(x: 3, y: 4) let d = Point(x: 2, y: 5) let cGreaterThanD = (c > d) // false let cLessThanD = (c < d) // false let cEqualToD = (c == d) // false As the above example demonstrates, the trouble arises in comparing two points when one point’s x and y properties are not both larger than the other point’s. In actuality, it is not reasonable to compare two points in this manner. Fix this problem by changing Point’s conformance to Comparable. Calculate each point’s Euclidean distance from the origin instead of comparing x and y values. This implementation should return true for a < b when a is closer to the origin than b. Use the formula shown in below Figure 25.1 to calculate a point’s Euclidean distance. Figure 25.1 Euclidean distance Euclidean distancePlease written by computer source 1) Write truth tables for the statement forms in A.∼p ∧ q B. p ∧ (q ∧ r) 2) Determine whether the statement forms are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence. 1. p ∨ (p ∧ q) and p 2. p ∨ t and t 3. (p ∧ q) ∧ r and p ∧ (q ∧ r) 4. (p ∧ q) ∨ r and p ∧ (q ∨ r) 3) Assume x is a particular real number and use De Morgan’s laws to write negations for the statements 1. −2 < x < 7 2. x < 2 or x > 5 3. 1 > x ≥ −3 4) Use truth tables to establish which of the statement forms are tautologies and which are contradictions. 1. (p ∧ q) ∨ (∼p ∨ (p ∧ ∼q)) 2. (p ∧ ∼q) ∧ (∼p ∨ q) 5) In the below, a logical equivalence is derived from Theorem 2.1.1. Supply a reason for each step. (p ∧ ∼q) ∨ (p ∧ q) ≡ p ∧ (∼q ∨ q) by (a) ≡ p ∧ (q ∨ ∼q) by (b) ≡ p ∧ t by (c) ≡ p by (d) Therefore, (p ∧ ∼q) ∨ (p ∧ q) ≡ p. 6) Use…
- 6. Consider the heuristics for this problem shown in the table above, h2 is admissible. true or false?1- Provide the following logical equivalences. (p↔q) ≡( p ∧ q )∧ ( q ∧ p ) (ii) (p∨ q) ≡ ( p ∧ q) ∧ (q ∧ p) 2- Test the validity of the following arguments If I drop out of school, I'll get a job at the bank. I'm not dropping out of school, so I'm not going to start a job at the bank. (ii) James is a cop or a football player. If he's a cop, he has a gun. James doesn't have a gun, so James is a football player 3- Prove by induction that n3 +2n is divisible by 3 with n > 0. 4- Prove that the product of any three consecutive integers is divisible by 6. 5- Prove that the sum of the squares of the first n positive integers is 6 n (n + 1) (2n + 1).Q.1.7 Discuss the features of a good test case.