ing sente 1- Who study mathematics every day? 2- Where do Ann work? 3- Universities must show that the principle is among it's core values. 4- Every house in the street have a garage. 5- Britians' coastline is very beautiful. 6- Greg is hers youngest daughter's husband. 7- Last year, I have one week's holiday. 8- Peter have no respect for other people's property.
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- In "Computing Machines and Intelligence," the English mathematician Alan Turing (1912-1954) wrote, "If each man had a definite set of rules of conduct by which he regulated his life, he would be a machine, but there are no such rules, so men cannot be machines." Using what you now know about Logic, determine whether this statement make sense and explain. Construct a truth table for Turing's statement and determine if this statement is a tautology.Determine whether the passage expresses an argument. Mark Yes or No. If it does express an argument, highlight the premise(s) (P) and underline the conclusion. 1. John must be in the library. I've checked the dorm, the cafeteria, and the game room, and I know he doesn't have a class now. 2. As TikTok video viewing has increased, the reading ability of public school children has decreased. Therefore, TikTok viewing contributes to reading problems. 3. If God is all good, he would not will evil. If he is all powerful, he would not permit evil. But there is evil. Therefore, God is not all powerful, or he is not all good. 4. If the Republicans win the next election, there will be a recession, federal revenues will fall, and the deficit will grow. If the Democrats win the next election, there will be increased spending and the deficit will grow. Since either the Republicans or the Democrats will win the next election, the deficit will grow. 5. Because I failed to study, I didn't pass the…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)
- Determine whether the following statements are logically equivalent or not. Show your work to clearly indicate your answer. p→¬qand¬(p→q)Prove ⊢ (¬A → A) → A in Hilbert deductive system. Note: In addition to the axioms and rule of inference of H, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook). You may not use theorem 3.31, as this is precisely that theorem.book Mordechai Ben-Ari Mathematical Logic for Computer Science Third Edition Prove {¬A} ⊢ (¬B → A) → B in Hilbert deductive system. Note: In addition to the axioms and rule of inference of H, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook). book Mordechai Ben-Ari Mathematical Logic for Computer Science Third Edition PLEASE solve these with the help of 3 axioms and 1 rule of inference with the derived proof 3.20-3.30.i) Carefully explain why p ↔ q can be written as “p is necessary and sufficient for q”. ii) Verify the second part of De Morgan’s laws, ∼ (p∨q) ≡ (∼ p) ∧(∼ q), in two ways: (a) Explain it using conversational English, using your own particular examples of statements for p and q. (b) Use a truth table.
- Determine which pairs of statements are equivalent.Note: You should be able to do this without a truth table.i. If I am happy, then the Giants won.ii. If the Giants win, then I am happy.iii. If the Giants lose, then I am unhappy.iv. If I am unhappy, then the Giants lose.Select all that apply:A. ii and ivB. i and ivC. iii and ivD. i and iiE. ii and iiiF. i and iiiThe king and the Chees board. A king is so impressed by one of his courtiers thet he offers to give him whatever he desires. The wily courtier replies that all he wants are some grains, as many as could be placed on each square of a chess board. Specially, grains are to be placed on each square of a chess board, as follows. One grain is to be placed in the first square,two in the second square , four in the third square and so on. (We'll assume the squares are big enough to accommodate the increasing numbers of grains they're each expected to hold). Assuming anNxN chess board, WAP in java that displays the exacy number of grains required to cover an NxN board. Input: integer N , Output:integer P (number of grains)The king and the Chees board. A king is so impressed by one of his courtiers thet he offers to give him whatever he desires. The wily courtier replies that all he wants are some grains, as many as could be placed on each square of a chess board. Specially, grains are to be placed on each square of a chess board, as follows. One grain is to be placed in the first square,two in the second square , four in the third square and so on. (We'll assume the squares are big enough to accommodate the increasing numbers of grains they're each expected to hold). Assuming anNxN chess board, WAP that displays the exacy number of grains required to cover an NxN board. Input: integer N , Output:integer P (number of grains)
- Suppose you wish to prove a theorem of the form “if p then q .”(a) If you give a direct proof, what do you assume and what do you prove?(b) If you give a proof by contraposition, what do you assume and what do you prove?(c) If you give a proof by contradiction, what do you assume and what do you prove?Problem 4) Translate the given statement into propositional logic using the propositions provided. (Do not simplify the expressions.) “It is not true that you can go out without a hat only if it is warm and not rainy.” Express your answer in terms ofh: “You can go out without a hat”, w: “It is warm”, andr: “It is rainy”.Solve the following puzzle by translating statements into logical expressions and reasoning fromthese expressions.Four friends have been identified as suspects for an unauthorized access into a computer system.They have made statements to the investigating authorities.1-Alice said Carlos did it.2-John said I did not do it.3-Carlos said Diana did it.4-Diana said Carlos lied when he said that I did it.If the authorities also know that exactly one of the four suspects is telling the truth, who didit?