**Logical Expressions of Quiz Show Participation**Let \( Q(x, y) \) be the statement "Student \( x \) has been a contestant on quiz show \( y \)." Express each sentence in terms of \( Q(x, y) \), quantifiers, and logical connectives, where the domain for \( x \) consists of all students at your school and for \( y \) consists of all quiz shows on television.**(a)** There is a student at your school who has been a contestant on a television quiz show.**(b)** No student at your school has ever been a contestant on a television quiz show.**(c)** There is a student at your school who has been a contestant on *Jeopardy!* and on *Wheel of Fortune*.**(d)** Every television quiz show has had a student from your school as a contestant.**(e)** At least two students from your school have been contestants on *Jeopardy!*

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Answered: 8. Let Q(x, y) be the statement…

Advanced Engineering Mathematics

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9. Let P(x) be the statement "x can speak Russian" and let Q(x) be the statement "x knows the computer language C++." Express each of these sentences in terms of P(x), Q(x), quantifiers, and logical connectives. The domain for quantifiers consists of all students at your school.
33. Rewrite each of these statements so that negations ap- pear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). x(x, y) a) xy(x, y) c)x(P(x, y) v Q(x, y)) d) (x³y-P(x, y) e)x(yz(x, y, z) z(x, y, z) b) x(x, y))
Express this in predicate logic: Each email address has exactly one email box. M(x): x is an email address B(x,y): x has an email box y
30. Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). (a)¬у³x²(x, y) (b) -VryP(x, y) (c) y(Q(y) AV-R(x, y)) (d)¬Jy(R(x, y) V VxS(x, y)) (e)y (VxzT(x, y, z) VrVzU (x, y, z)) 7
8. Use quantifiers and predicates with more than one variable to express this statement: “There a person in your class who has taken every math class at Liberty University." Let the domain of x be all people and the domain of y be all classes.
Apply De Morgan's law to the following expression to obtain an equivalent expression in which each negation sign applies directly to a predicate. (P(x) ^ (Q(x) v R(x))) O 3x (-P(x) v (¬Q(x) ^ ¬R(x))) -3x P(x) or 3x ¬(P(x) ^ Q(x)) O 3x (P(x) v (Q(x) ^ ¬R(x))) O None are equivalent.
21. Let the following predicates be given. The domain is all computer science classes. I(r) = "x is interesting." U(x) = "x is useful." H(x, y) = “x is harder than y." M(x, y) = “x has more students than y." (a) Write the following statements in predicate logic. i. All interesting CS classes are useful. ii. There are some useful CS classes that are not interesting. iii. Every interesting CS class has more students than any non- interesting CS class. (b) Write the following predicate logic statement in everyday English. Don't just give a word-for-word translation; your sentence should make sense. (3)[I(x) A (Vy)(H(x,y)→ M(y, x))] (c) Formally negate the statement from part (b). Simplify your negation so that no quantifier lies within the scope of a negation. State which derivation rules you are using. (d) Give a translation of your negated statement in everyday English.
1. Let Q(x, y) be the statement "Student x has been a contestant on quiz show y." Express in the box below this instruction each of these sentences in terms of Q(x, y), quantifiers, and logical connectives, where the domain for x consists of all students at your school and for y consists of all quiz shows on television. a) There is a student at your school who has been a contestant on a television quiz show. b) No student at your school has ever been a contestant on a television quiz show. c) Every television quiz show has had a student from your school as a contestant. You may copy the sign , A, V,, , E, 31, 3,

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