GO 2023 Topic Wise Free Test 2 | First Order Logic | Question: 13

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GO Classes asked in Mathematical Logic Dec 13, 2022 edited Dec 13, 2022 by Lakshman Bhaiya

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Which of the following formulas does not express that there is exactly one element of $\text{D}$ that has property $\text{P}(x)?$

$\exists x \in \text{D} \forall y \in \text{D}[\text{P}(y) \leftrightarrow y=x]$
$\exists x \in \text{D}[\text{P}(x) \wedge \forall y \in \text{D}[y \neq x \rightarrow \neg \text{P}(y)]]$
$\exists x \in \text{D} \text{P}(x) \wedge \forall x_1, x_2 \in \text{D}\left[\text{P}\left(x_1\right) \wedge \text{P}\left(x_2\right) \rightarrow x_1=x_2\right]$
$\exists x \in \text{D} \forall y \in \text{D}[\text{P}(y) \rightarrow y=x]$

GO Classes asked in Mathematical Logic Dec 13, 2022 edited Dec 13, 2022 by Lakshman Bhaiya

166 views

2 Answers

Udhay_Brahmi · Answer 1 · 2022-12-27T07:57:11+0000

Reason why Option D is the correct option :

Consider a scenario where (x1,x2,x3) satisfies property P.

Now, If we run the option D, then (for x1 we get y = x1), (for x2 we get y = x2), (for x3 we get y = x3).

So, option D tell → “Its okay to have multiple x satisfying P(x)” which is not what we want.

Good question.✨