GO Classes CS Test Series 2025 | Discrete Mathematics | Topic Wise Test 1 | Question: 13

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GO Classes CS Test Series 2025 | Discrete Mathematics | Topic Wise Test 1 | Question: 13

GO Classes asked Apr 14, 2022 • retagged 4 days ago by Lakshman Bhaiya

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Consider the following predicates.

$\text{Rabbit}(x) = x$ is a rabbit.
$\text{Cute}(x) = x$ is cute.

Consider the following statement $\text{E},$ where the domain of every variable is set of all animals in a jungle $\text{J}.$
$\text{E} = \exists x ( \text{Rabbit}(x) \rightarrow \text{Cute}(x) )$

If statement $\text{E}$ is false, then which of the following is necessarily true?

There is no animal other than rabbits in the jungle $\text{J}.$
There is no cute animal in the jungle $\text{J}.$
There is no cute rabbit in jungle $\text{J}.$
There is some rabbit who is not cute in jungle $\text{J}.$

GO Classes asked Apr 14, 2022 • retagged 4 days ago by Lakshman Bhaiya

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John_Smith

commented Nov 30, 2023

i

edited

by John_Smith Nov 30, 2023

@Deepak Poonia Sir, I understand that we should not consider the possibility of the domain being empty while computing the answer, unless the question explicitly specifies it, but, as a variation of this question, what if the question indeed mentioned that the domain may or may not be empty, i.e., there may or may not be any animals in the jungle $J$. Then how do we even proceed? IIUC, it feels like we encounter the same kind of ambiguities as arguing whether or not $\phi \subseteq \phi$ (where $\phi$ denotes an empty set), i.e., for any single-valued predicate $P(x)$, $[\forall x \in \phi, \ P(x)]$ (and $[\exists x \in \phi, \ P(x)]$) seem to be both TRUE and FALSE.

P. S. I also made a related comment here.

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@Deepak Poonia Sir, I understand that we should not consider the possibility of the domain being empty while computing the answer, unless the question explicitly specifies it, but, as a variation of this question, what if the question indeed mentioned that the domain may or may not be empty, i.e., there may or may not be any animals in the jungle $J$. Then how do we even proceed? IIUC, it feels like we encounter the same kind of ambiguities as arguing whether or not $\phi \subseteq \phi$ (where $\phi$ denotes an empty set), i.e., for any single-valued predicate $P(x)$, $[\forall x \in \phi, \ P(x)]$ (and $[\exists x \in \phi, \ P(x)]$) seem to be both TRUE and FALSE.
P. S. I also made a related comment here. — John_Smith, Nov 30, 2023
@John_Smith, we don’t consider domain to be empty in mathematical logic because if we allow domain to be empty by default, we may get many non-intuitive results.

For example, $\forall x P(x) \rightarrow \exists x P(x)$ is a valid (always true for every predicate $P(x)$ & for every domain) statement & it is quite intuitive that it must be valid(i.e. always true). But if we allow empty domain by default, then this statement becomes invalid.

As a corner case, if it is given that the domain is empty, then $\forall x P(x) $ is taken to be true and $ \exists x P(x)$ is taken to be false.

So, $[ \forall x \in \phi, P(x) ] = True$ ; $[ \exists x \in \phi, P(x) ] = False.$ — Deepak Poonia, Nov 30, 2023
@Deepak Poonia Sir, I see. So, the corner cases are more like conventions rather than strict mathematical properties or theorems. Got it. Thank you so much for the help. — John_Smith, Dec 1, 2023
$\exists x(Rabbit(x)\rightarrow Cute(x))\equiv \exists x(\neg Rabbit(x)\vee Cute(x))\equiv False$ (Given)

$\exists x(\neg Rabbit(x)\vee Cute(x))$ is false when for each element in domain it is False and since it contains disjunction both $\neg Rabbit(x)=False$ and also $Cute(x)= False$.

So, for each element in domain $Rabbit(x)=True$ and $Cute(x)=False$ — Kiyoshi, May 10, 2023
@Kiyoshi , “interpretation” is the wrong word to be used here.

You meant “element of domain” instead of “interpretation” in all places. — Deepak Poonia, May 10, 2023

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