Let F(x,y) be the statement “x likes y" where the domain consists of all people in the world. (a). Use quantifiers to express "Not everyone likes Jack and Jill". (b). Give an equivalent statement of (a) using quantifiers.
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- 4. 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)Suppose P(a) is the predicate "a is prime" and Q(a) is the predicate "a is divisible by 3", for all integers a. In the following, the domain is the set of integers. What is the truth value of ∃x(P(x)∧Q(x))? Justify your answer.I. Let P (x) be the statement “2x = x2 .” If the domain consists of the integers, what are the truth values? a) P(0) b) P(1) c) P(2) d) P(−1) e) ∃x P(x) f ) ∀x P(x) 2. Let Q(x) be the statement “x > x - 1.” If the domain consists of all integers, what are the truth values? a) Q(10) b) Q(−2) c) Q(999) d) ∃x Q(x) e) ∀x Q(x) f) ∃x ˺Q(x) g) ∀x ˺Q(x)
- Let l be a line in the x-yplane. If l is a vertical line, its equation is x = a for some real number a. Suppose l is not a vertical line and its slope is m. Then the equation of l is y = mx + b, where b is the y-intercept. If l passes through the point (x₀, y₀), the equation of l can be written as y - y₀ = m(x - x₀). If (x₁, y₁) and (x₂, y₂) are two points in the x-y plane and x₁ ≠ x₂, the slope of line passing through these points is m = (y₂ - y₁)/(x₂ - x₁). Instructions Write a program that prompts the user for two points in the x-y plane. Input should be entered in the following order: Input x₁ Input y₁ Input x₂Please answer the following question in depth with full detail. Consider the 8-puzzle that we discussed in class. Suppose we define a new heuristic function h3 which is the average of h1 and h2, and another heuristic function h4 which is the sum of h1 and h2. That is, for every state s ∈ S: h3(s) =h1(s) + h2(s) 2 h4(s) =h1(s) + h2(s) where h1 and h2 are defined as “the number of misplaced tiles”, and “the sum of the distances of the tiles from their goal positions”, respectively. Are h3 and h4 admissible? If admissible, compare their dominance with respect to h1 and h2, if not, provide a counterexample, i.e. a puzzle configuration where dominance does not hold.You and your friends decided to hold a “Secret Santa” gift exchange, where each person buys a gift for someone else. To see how this whole thing works, let’s consider the following example. Suppose there are 7 people A, B, C, D, E, F, and G. We denote x → y to mean “x gives a gift to y.” If the gift exchange starts with person A, then they give a gift to E. Then E gives a gift to B. And it is entirely possible that B gives a gift to A; in such a case we have completed a “cycle.” In case a cycle occurs, the gift exchange resumes with another person that hasn’t given their gift yet. If the gift exchange resumes with person D, then they give a gift to G. Then G gives a gift to F. Then F gives a gift to C. Then finally C gives a gift to D, which completes another cycle. Since all of the people have given their gifts, the giftexchange is done, otherwise the gift exchange resumes again with another person. All in all, there are two cycles that occurred during the gift exchange: A → E → B → A…
- What is the truth value of each of the following wffs in the interpretation where the domain consists of the integers, A(x)A(x) is “x<5x” and B(x)B(x) is “x<7 ”? Group of answer choices a. (∀x)[B(x)→A(x)] b.(∃x)[A(x)∧B(x)] c.(∀x)[A(x)→B(x)] d.(∃x)A(x)Note that for this question, you can in addition use ``land'' for the symbol ∧ ``lor'' for the symbol ∨ ``lnot'' for the symbol ¬. Given the following three sentences:A) Every mathematician is married to an engineer.B) A bachelor is not married to anyone.C) If George is a mathematician, then he is not a bachelor. a) Convert A,B,C into three FOL sentences, whereMn(x): x is a mathematician.Er(x): x is an engineer.Md(x,y): x is married to y.Br(x): x is a bachelor.george: George is a constant. b) Show that A does-not-entail C. (Hint: Consider defining an interpretation I such that I models A, but does-not-model C.)c) Show that {A,B} entails C. (Hint: For a given interpretation I, consider two difference cases, the case where Mn(george) is true, and the case Mn(george) is false. For both cases, argue that it is always that I models C).d) Convert A,B, lnot C into a set of clausal forms, number your clauses. (Note that C is negated here!) e) Derive the empty clause from the set of clauses…Rewrite the following proposition as unambiguous English sentences. The relevant predicates are defined as follows: Again, let’s let P be a set of all people A(x) means “x is teaching the AI course” T(x) means “x is taking AI course” F(x) means “x has a twitter account.” C(x) means “x likes to read.” For example, the proposition ∃x { P [A(x)∧ C(x)]} could be translated as “There is a person who is teaching and likes to read” P [T(x) → (F(x) C(x))] ?
- 2) (L2) Prove using laws of logic that the conditional proposition (p ∧ q) → r is equivalent to (p ∧ ¬ r) →¬ q. 3) (L3) Show that the converse of a conditional proposition p: q → r is equivalent to the inverse of proposition p using a truth table. 4.1) (L4) Show whether ((p ∧ (p→q)) ↔ ¬p) is a tautology or not. Use a truth table and be specific about which row(s)/column(s) of the truth table justify your answer. 4.2) (L4) Give truth values for the propositional variables that cause the two expressions to have different truth values. For example, given p ∨ q and p ⊕ q, the correct answer would be p = q = T, because when p and q are both true, p ∨ q is true but p ⊕ q is false. Note that there may be more than one correct answer. r ∧ (p ∨ q) (r ∧ p) ∨ qWe have shown how to use truth tables to determine if two formulas are truth-functionally equivalent.If two formulas F and G are truth-functionally equivalent we introduce another symbol ↔, aptly called thebiconditional. Here is the truth table for the biconditional.p q (p ↔ q)1 1 11 0 00 1 00 0 1Now we shall say that F and G are truth-functionally equivalent if (F ↔ G) is a tautology.There are other properties of two formulas that we are usually interested in besides truth-functionalequivalence. One of these properties is when two formulas are mutually exclusive. We say two formulas Fand G are mutually exclusive if (F ∧ G) is contradictory (unsatisfiable).Now using truth tables determine whether the following formulas are truth-functionally equivalent or mutuallyexclusive.(a) p and ¬p (b) p and ¬¬p (c) ¬(p ∧ ¬q) and (p → q) (d) (¬p ∨ q) and (p → q) (e) ¬(¬p ∨ ¬q) and (p ∧ q) (f) ¬(¬p ∧ ¬q) and (p ∨ q)Let P(x) and Q(x) be predicates and suppose D is the domain of x. For the statement forms in the given pair, determine whether they have the same truth value for every choice of P(x), Q(x), and D, or whether there is a choice of P(x), Q(x), and D for which they have opposite truth values. ∃x ∈ D, (P(x) ∧ Q(x)) and (∃x ∈ D, P(x)) ∧ (∃x ∈ D, Q(x))