Let Q(x, y) be the statement “student x has been a contestant on quiz show y.” Express 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) 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.
(Reference: Discrete Mathematics (7th Edition) Kenneth H. Rosen, Exercise-1.5 Question No.-8)
I am not able to solve this question. What the approach to solve this question? Please Explain
I hope that you are clear with the quantifiers and their use. I will show in this answer how to use them together with the logical connectives in order solve the problem.
a) Given statement is: "There is a student in your school who has been a contestant in a television quiz show". First, identify the set. Clearly it is the set of students in the school. Name that set $S$. What does the statement say? It says "there is a student", which in terms of quantifiers means $\exists x \in S$. What property does this $x$ hold? It has contested in a television quiz show. So, for us, $y = $ television quiz show. Now, we combine all this information in a single logic statement involving quantifiers:-
$$\exists x \in S \left( Q \left( x, \text{ television quiz show} \right) \right)$$
b) Given statement is: "No student at your school has ever been a contestant on a television quiz show". This can also be said as, "Given any student in your school, the student has not been on a television quiz show". So, the quantifier being used is $\forall$. What property does every student hold? It is that the student has not participated in television quiz show. If $Q \left( x, \text{ television quiz show} \right)$ means that $x$ has participated in the show, then $\neg Q \left( x, \text{ television quiz show} \right)$ means that $x$ has not participated. Hence, the given statement in form of logical connectives is
$$\forall x \in S \left( \neg Q \left( x, \text{ television quiz show} \right) \right)$$
c) This goes same as a) with two television shows thereby making the logical statement as:
$$\exists x \in S \left( Q \left( x, \text{Jeopardy} \right) \wedge Q \left( x, \text{Wheel of Fortune} \right) \right)$$
d) Given statement: "Every television show has had a student from your school as a contestant." Here, two sets are involved: The set of all television shows, say $T$, and second, the set of all students of your school, say $S$. Now, first we pick "any" television show, i.e., $\forall t \in T$. What property does this $t$ hold? It holds that one student from your school, i.e., $\exists s \in S$ has participated in it, i.e., $Q \left( s, t \right)$ holds true. Hence, the given statement in terms of quantifiers is given as
$$\forall t \in T \left( \exists s \in S \left( Q \left( s, t \right) \right) \right)$$
e) Given statement is: "At least two students have been contestants on Jeopardy". Since the statement says, there are "at least two" students, we use the quantifier $\exists x \in S$ and $\exists y \in S$ with an additional property that $x \neq y$ (so that at least two holds in general). The property that both the students hold is that for both of them $Q \left( x, \text{Jeopardy} \right)$ and $Q \left( y, \text{Jeopardy} \right)$ holds true. So, in terms of quantifiers, we write:
$$\exists x \in S \wedge \exists y \in S \left( x \neq y \wedge Q \left( x, \text{Jeopardy} \right) \wedge Q \left( y, \text{Jeopardy} \right) \right)$$