QUESTION 2 Let F(x.y): "x is friend with y", N(x): "x is funny", W(x): "x is wise", R(x): "x is fair". The domain for x and y is the set of all persons. Select the symbolic (s) of the following senetnce: "Some fair persons are friends with Harry" O vx R(X) A F(x,Harry) Vx R(X) -F(x,Harry) None of all the proposed answers 3x R(X) A F(x,Harry) O 3x R(x)-F(x,Harry)
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- Consider the following predicates defined on N + .E(n) denotes “n is even”, and P (n) denotes “n is prime”. (a) Translate the following into ordinary English.i. ∃n (P (n) ∧ E(n)).ii. ∀n (E(n) ∨ ¬P (n)).iii. ¬∀n (E(n) ∨ P (n)).1. Big-O Notation Let f and g be functions from the set of integers or the set of real numbers to the set of real numbers. We say that f ( x ) is O ( g ( x ) ), read as "f ( x ) is big-oh of g ( x )", if there are constants C and k such that | f ( x ) | ≤ C | g ( x ) | whenever x > k. (a) Show that f(x) = x2 + 2x + 1 is O(x2) Solution: When x>1; ? 2 (1 + 2 ? + 1 ? 2 ) < ? 2 (1 + 2 1 + 1 1 2 ) = 4? 2 So, ??? ? > 1, ? 2 + 2? + 1 < 4? 2 From the definition 0 ≤ f(x) ≤ cg(x) for x≥1 Hence, for N0 = 1; c=4; and g(x)=x2 for N0 = 2; c=3; and g(x)=x2 for N0 = 3; c=2; and g(x)=x2 … Therefore, ? ? + ?? + ? = ?(? ? ) O(g(x)) = {f(x)|there exist positive constant c and N0 such that 0 ≤ f(x) ≤ cg(x) for all x≥N0} 2. Show that 7x2 is O(x3). 3. Suppose there are x number of boxes to be delivered to x number of household that is 2km apart, what is the distance travelled by the transport delivery service? 4. In number 3, suppose that each boxes…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.
- ewrite the following proposition as unambiguous English sentences. The relevant redicates are defined as follows: Again, let's let P be a set of all people a. A(x) means "x is teaching the Al course" b. T(x) means "x is taking Al course" c. F(x) means "x has a twitter account." d. C(x) means "x likes to read."Q2 – Proof of Correctness In this question, you will use strong induction to prove that your new algorithm works correctly. In other words, you will prove that nN xR-{0} FP(x,n) = xn a) Predicate Function Your conjecture has already been stated in symbolic form: It is a statement of the form nN, P(n) What is the predicate function P(n)? b) Proof: Base cases c) Proof: Inductive step setup This is the beginning of the inductive step where you are stating the assumptions in the inductive step and what you will be proving in that step. As you do so, identify the inductive hypothesis. d) proof: inductive stepsSuppose team A plays in a best-of-7-game series. In sports sometimes teams play in a series until one team has won 4 games in the series. When a team wins its fourth game the series concludes because after a team has won 4 games it is guaranteed to win more games than its opponent even if all 7 games are played. When team A wins a game we write a W and when team A loses a game we write a L. So the sequence WWLLLWL means team A won the first, second and sixth games in the series and lost the other games. Team A lost the series. The sequence LWWLWW indicates team A won the second, third, fifth and sixth games. Team A won the series in 6 games. Given that team A wins the fifth game how many sequences of wins and losses are possible if team A wins the series?
- QUESTION 5 Let P(x; y) : x plays in y A(x) : x is athletic. S(x) :x is smart. E(y) : y is in English league. F(y) : y is famous. Assume the domain of x is all players and the domain of y is all football teams. The symbolic of the following sentence : "All smart and athletic players play in some English league teams" 1. for all x there exists y space S left parenthesis x right parenthesis logical and A left parenthesis x right parenthesis logical and E left parenthesis y right parenthesis logical and P left parenthesis x comma y right parenthesis 2. None of all the proposed answers 3. for all x there exists y space S left parenthesis x right parenthesis logical and A left parenthesis x right parenthesis rightwards arrow E left parenthesis y right parenthesis logical and P left parenthesis x comma y right parenthesis 4. for all x there exists y space S left parenthesis x right parenthesis logical and A left parenthesis x right parenthesis logical or E left…(1) Formalize the predicate A(s, t), “s is alphabetically prior to t,” wheres and t are words of length 5, i.e., functions of type {0, 1, 2, 3, 4} →{a, b, c, . . . , z}. You can use the symbol < on both numbers and alphabetcharacters, so, e.g., a < b < c < ···< z. (2)Formalize the predicate EvC (f ), “f is eventually constant.”Write a recursive function in f#, named specialSum, that has the followeing signature: int * int -> int, where sum(m,n) = m + (m +1) + (m+2) + ... + (m + (n-1)) + (m+n) for m >= 0 and n >= 0: (Hint use two clauses with (m,0) and (m,n) A PATTERNS.) start code with let rec specialSum (m,n) match m,n with | m,0 -> | m,n - >
- Code in Python The function first_three takes one string parameter, word, and returns a slice of length three from the front of word. However, if word has length less than three, then the None object is returned instead. Hint: Use an if-else decision and the slicing operator [ : ] to solve this problem. For example: Test Result word = "happy" if type(first_three(word)) != type("ok"): print("error") else: print(first_three(word)) hap word = "dog" if type(first_three(word)) != type("ok"): print("error") else: print(first_three(word)) dog word = "" if type(first_three(word)) != type(None): print("error") else: print(first_three(word)) Nonecode in scala def unfold[A, S](z: S)(f: S => Option[(A, S)]): LazyList[A] = f(z) match { case Some((h, s)) => h #:: unfold(s)(f) case None => LazyList() } [Note: #:: is the Cons constructor of LazyLists] A triple (x, y, z) of positive integers is pythagorean if x2 + y2 = z2. Using the functions studied in class, define a function pyth which returns the list of all pythagorean triples whose components are at most a given limit. For example, function call pyth(10) should return [(3, 4, 5), (4, 3, 5), (6, 8, 10), (8, 6, 10)]. [Hint: One way to do this is to construct a list of all triples (use unfold to create a list of integers, and then a for-comprehension to create a list of all triples), and then select the pythagorean ones.If f is injective, while both f ◦ g and f ◦ g0 are defined, show that f ◦ g = f ◦ g0 =⇒ g = g0.