GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 29

Question

Which of the following is correct formalization of the sentence

"There is a student who is loved by every other student."?

Here, $\text{Loves}(p,q)$ means $p \text{ loves } q.$

• A. $\exists x(\text{Student}(x) \wedge \forall y(\text{Student}(y) \rightarrow \text{Loves}(y,x)))$
• B. $\exists x(\text{Student}(x) \wedge \forall y(\text{Student}(y) \wedge \neg(y=x) \rightarrow \text{Loves}(y,x)))$
• C. $\exists x(\text{Student}(x) \rightarrow \forall y(\text{Student}(y) \wedge \text{Loves}(y,x)))$
• D. $\exists x(\text{Student}(x) \rightarrow \forall y(\text{Student}(y) \wedge \neg(y=x) \wedge \text{Loves}(y,x)))$

Comments

form the question how do i know student doesn’t love himself.

eg raju doesn’t loves raju??

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@amit_prasad
They stated every other student, which indirectly means all students except himself.
If they would have said every student, then your point would have been correct.

Answer 1

"There is a student who is loved by every other student."

Here, $\text{Loves}(p,q)$ means $p \text{ loves}\; q.$

We can interpret in this way,

• $\text{Raju}$ is loved by $\text{Rani}$
• $\text{Raju}$ is loved by $\text{Vani}$
• $\text{Raju}$ is loved by $\text{Rakesh}$
• $\text{Raju}$ is loved by $\text{Suresh}$
• and, so on..

In other way,

• $\text{Rani}$ loves $\text{Raju}$
• $\text{Vani}$ loves $\text{Raju}$
• $\text{Rakesh}$ loves $\text{Raju}$
• $\text{Suresh}$ loves $\text{Raju}$
• But, Raju cannot loves himself.

Similarly, we can write like this,

• $\text{Loves(Rani,Raju)}$
• $\text{Loves(Vani,Raju)}$
• $\text{Loves(Rakesh,Raju)}$
• $\text{Loves(Suresh,Raju)}$

Now, Lets take $\text{Student}(x):x$ is a student$,\text{Student}(y):y$ is a student.

Domain of discourse:All the people

• There is a student $x,$ who loved by student $y_{1}$
• There is a student $x,$ who loved by student $y_{2}$
• There is a student $x,$ who loved by student $y_{3}$
• ----------------------------------------------------------------------
• There is a student $x,$ who loved by student $y_{n}$

Now, we can represent above statements in this way,

• $y_{1} \;\text{loves} \;x$
• $y_{2} \;\text{loves} \;x$
• $y_{3} \;\text{loves} \;x$
• -------------------------------
• $y_{n} \;\text{loves} \;x$

We can write above statements, like this

• $\text{Loves}(y_{1},x)$
• $\text{Loves}(y_{2},x)$
• $\text{Loves}(y_{3},x)$
• ------------------------------
• $\text{Loves}(y_{n},x)$

Now, write above statements in proper way,

There exist at-least one student $x,$ who loved by $y,$ if $y$ is a student, and $x$ can not loved by himself

$\exists x(\text{Student}(x) \wedge \forall y(\text{Student}(y) \wedge \neg(y=x) \rightarrow \text{Loves}(y,x)))$

So, the correct answer is $(B).$

Comments

beaitul. I chose option A. now got it

raju cannot love raju

Answer 2

Option A here represent "some student is loved by every student can includes himself" where as option B represent "There is a student who is loved by every other student".

Options C and D are true if a non student is present and hence wrong.