Logic basic

Question

Let Q(x,y) be the statement "Student x has been a contestant on quiz show y "
Express each of these statement in terms of Q(x,y) , quantifiers and logical connectives

1)There is a student at your school who has been a contestant on a televison quiz show

2) No student at your school has ever been a contestant ona TV quiz show

3)There is a student at your school who has been a contestant on jeopardy and Wheel of fortune

4)Atleast 2 studnet from your school have been contestant on Jeopardy

Answer 1

Please check and point my mistakes :)

1)There is a student at your school who has been a contestant on a televison quiz show

∃!x∃!y Q(x,y)

2) No student at your school has ever been a contestant on a TV quiz show =

∼ (∀x∀y Q(x,y))

3)There is a student at your school who has been a contestant on jeopardy and Wheel of fortune =

∃!x(Q(x,Jeopardy )⋀ Q(x,Wheel of fortune))

4)Atleast 2 studnet from your school have been contestant on Jeopardy

∃x∃y((x!=y) ⋀ Q(x,Jeopardy )⋀Q(y,Jeopardy ))

Comments

2) Try digging the negation deeper into the statement. ヨstudヨquizㄱQ(stud, quiz). What does that mean? It means there is at least a student quiz pair for which the student has not participated in the quiz. In other words, it is not the case that every student in the domain has participated in every quiz comp. At least one non participating pair is enough to make the statement given by you true. Is that what was asked in the question? No. Correct answer should be ㄱヨstudヨquiz Q(stud, quiz).

Plus, I'm not quite sure about your usage of uniqueness quantifier in the answers. Say for question 1, What if two students participated in two quiz comps? Will that make the statement false? Even then there is 'a' student who have participated in 'a' quiz competition. The question doesn't say 'one and only one'. Existential quant seems more appropriate there.

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2.  ∀x∀y ∼Q(x,y)