Let p be “It is cold” and let q be “It is raining”. Give a simple verbal sentence which describes each of the following statements: a) ~p b) p ∧ q c) p ∨ q
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Let p be “It is cold” and let q be “It is raining”. Give a simple verbal sentence which describes each of the following statements:
a) ~p
b) p ∧ q
c) p ∨ q
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- Let p, q, and r represent the following statements: p: The first ball is red. q: The second ball is white. r: The third ball is blue. Which would be the correct English statement for the symbolic form (¬¬p ∧∧ q) →→ r ? Group of answer choices A .If the first ball is red or the second ball is white, then the third ball is blue. B. If the first ball is not red or the second ball is white, then the third ball is blue. C .If the first ball is not red and the second ball is white, then the third ball is blue. D. If the first ball is red and the second ball is white, then the third ball is blue.6. Given, A = p A (p V q) and B = p v (p A q). State whether A = B or not. 7. Let P(x), Q(x), and R(x) be the statements. "x is a student," “x is smart," and "x is shy," respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), and R(x), where the domain consists of all people. a. Some students are shy. b. All smart people are not shy.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…
- Consider statements p and q. . p: Mary is practicing golf. q: Aldo is dancing. For parts (a) and (b), fill in the symbolic form. For part (c), choose the descriptive form. Descriptive form Symbolic form (a) Mary is not practicing golf but Aldo is dancing. (b) It is not the case that "Aldo is dancing or Mary is practicing golf". | (c) (Choose one) ~qv~pSuppose that p and q are statements so that p → q is false. Find the truth values of each of the following. (a) ~p → q TrueFalse (b) p ∨ q TrueFalse (c) q → p TrueFalse1 Let . Let p and q be the propositions “The election is decided” and “The votes have been counted,” respectively. Express each of these compound propositions as an English sentence. A. ¬p → ¬qB. p ∧ qC. ¬p ∨ qD. ¬p ∧ (p ∨ ¬q)E. p ↔ ¬qF. ¬p → ¬q
- Suppose P and Q are the statements: P: Jack passed math. Q: Jill passed math. (a) Translate “Jack and Jill both passed math” into symbols. (b) Translate “If Jack passed math, then Jill did not” into symbols. (c) Translate “P ∨ Q” into English. (d) Translate “¬(P ∧ Q) → Q” into English.1. Mathematical logic is a subfield that considers the analysis of logical propositions for the purpose of verifying whether a statement is true or false. Furthermore, propositions can be simple or compound. In the case of compounds, they must be joined with logical operators. Consider the following simple and compound propositions below. Also consider the “>” sign meaning “greater than”. ( ) It is not true that 11 is an even number ( ) It is not true that 11 is a prime number ( ) (2 + 4 = 6) and (1 > 3) ( ) (2 + 2 = 4) ou (3 + 3 = 7) ( ) Brasília é a capital do Brasil ou 2² = 5 Mark the alternative that shows the sequence of logical values, in order, of each proposition. Consider “T” as “true” and “F” as “false”. A V – F – F – V – V B F – V – F – V – F C V – F – V – F - V D F – V – V – V - F E V – V – F – V - FLet D = {-1, -2, -3} and E = {−3, 1, 2, 3, 5}. Write negations for each of the following statements and determine if the given statement is true or its negation. Explain your answer. (i) V E D, ³y € E such that x + y = 0. (ii) ay € E such that vx € D, x + y ≥ 2. (iii) vy € E, 3x = D such that xy ≥ 0.
- Q: The propositional variables b, v, and s represent the propositions: b: Alice rode her bike today. v: Alice overslept today. s: It is sunny today. Select the logical expression that represents the statement: “Alice rode her bike today only if it was sunny today and she did not oversleep.” b→(s→¬v) 2. b→(s∧¬v) 3. s∧(¬v→b) 4. (s∧¬v)→b Group of answer choices b→(s∧¬v) b→(s→¬v) s∧(¬v→b) (s∧¬v)→b 2); The following two statements are logically equivalent (p → q) ∧ (r → q) and (p ∧ r) → q Group of answer choices True FalseQ: The propositional variables b, v, and s represent the propositions: b: Alice rode her bike today. v: Alice overslept today. s: It is sunny today. Select the logical expression that represents the statement: “Alice rode her bike today only if it was sunny today and she did not oversleep.” b→(s→¬v) 2. b→(s∧¬v) 3. s∧(¬v→b) 4. (s∧¬v)→b Group of answer choices A): b→(s∧¬v) B): b→(s→¬v) C): s∧(¬v→b) D): (s∧¬v)→bConsider the following true propositions:• p : The applicant has passed the learner permit test.• q : The applicant has passed the road test.• r : The applicant is allowed a driver’s license.For each of the following sentences write,symbolically, the compound proposition that correspondstothe given sentence in English asit is written (do not change the order or form of the expression).a) The applicant did not pass the road test but passed the learner permit test.b) If the applicant passes the learner permit test and the road test, then the applicant is allowed adriver’s license.c) Passing the learner permit test and the road test are necessary for being allowed a driver’slicense.d) The applicant passed either the learner permit test or the road test, but not both.e) It is not true that the applicant does not pass the road test and is allowed a driver’s license.