Let p, q and r be propositions: p: Jasper has sore eyes q: Jasper misses the last day of the board exam r: Jasper passes the board exam Write the following propositions in terms of the given connectives: a.) q→¬r b.) q ↔ p c.) p → q ⋀ ¬r

Consider the following predicates.  P(x) is the statement "x−4<1".Q(x) is the statement "x+4⩽3".R(x) is the statement "x−4>0". Determine the truth value of the following propositions.  P(8)→¬Q(−2) T or F? Q(3)∧¬R(3) T or F? R(5)→P(5) T or F?

Please Help with the question below: Suppose you are given some facts in First-Order Logics: a) Andi is a professor b) All professors are people. c) Ani is the dean. d) All Deans are professors. e) All professors consider the dean a friend or don’t know him. f) Everyone is a friend of someone. g) People only criticize people that are not their friends. h) Andi criticized Ani. Prove that: Ani is not Andi’s friend.

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The domain is the set {a, b, c, d, e}. The table below shows the value of two predicates for each of the elements in the domain. Ex: The value of P(c) is false because the truth value in row c and column P is F. PQ a FT bFF CFF d TF e F F Check Next Give exactly one counterexample that shows the proposition VxP(x) is false. Ex: d Give exactly one example that shows the proposition 3xQ(x) is true. 2

Convert each of the following argu- ments into formal statements, e.g., define sentences existentially and/or universally quantified statements. Then determine which rules of logic have been applied and explain whether or not they have been ap- plied correctly. if you do the homework you will pass the final. Somebody did not pass the final. Therefore somebody did not not do the homework. If you do the homework you will pass the final. Curly did not do the homework. Therefore Curly will not pass the final. If you don’t do the homework, you won’t pass the final. Moe did the homework. Therefore Moe passed the final.

Need the right answer among choices and also an explanation of the answer.

Select ALL the sentences of FOL. G(P) AxLikes(x,pia) G(p)  Pia   AxLikes(x,y)   AxGuilty(x)  Guilty(x)

Let p, q, and r be the propositions p: You started a new video game this week. q: You finished all its quests.r: You didn’t win the platinum trophy.Express each of these propositions as an English sentence.¬p → ¬q(p ∧ q) → (¬r)((¬p) → (¬q)) ∨ (q → (¬r))(p ∧ q) ∨ (¬q ∧ r)

3. Express the following English sentences using propositional functions, quantifiers and logical connections. (a) Someone in the class is helped by everyone in the class. (b) There is a hard working student in the class who has not been helped by any student in class. (c) For everyone in the class, someone who is smart has helped him/her. SO (d) There are two people in the class who have not helped each other. (e) There is a hard working student in the class who is helped by every smart student in class. (f) There are some companies that hire only hard working students.