Awesome q2a theme
0 votes
17 views

Consider the following first order logic statements:

I: $\forall_x\forall_y p(x,y)$

II: $\forall_x\exists_y p(x,y)$

III: $\exists_y\exists_x p(x,y)$

IV: $\exists_y\forall_x p(x,y)$

 

Which of the following is not true about above statements?

(A) if is true then II, III, IV are true.

(B) if II is true then III, IV are true.

(C) is IV is true then II, III are true.

(D) None of these.

 

Ans given is (B).

Now i think that statement of (A) is also false (and can be answer too) because if domain of $x$ and $y$ is empty then statement will trivially be true. But in this case III and IV are false. So options (A) will be false.

Same for (C) if domain of $x$ is empty then IV will be true for all $y$ but in this case III will be false. So, options C is also false and hence answer too.

Is my argument correct or not?

in Mathematical Logic by (387 points) | 17 views
+1

Not sure about the argument. Let’s see what others comment. But just a little help from my side.

$\\ \forall x \forall y\leftrightarrow \forall y \forall x\\ \forall x \forall y \rightarrow \exists y \forall x\\ \exists y \forall x\rightarrow \forall x \exists y\\ \forall x \exists y\rightarrow \exists y \exists x\\ \exists y \exists x \leftrightarrow \exists x \exists y\\ \\ \forall y \forall x\rightarrow \exists x \forall y\\ \exists x \forall y\rightarrow \forall y \exists x\\ \forall y \exists x\rightarrow \exists x \exists y$

either directly observe from this or if you write all this implications and bi-implications in a systematic order you will observe a figure just like a hexagon which will indicate that B is not true

0

I'm thankful to you for your efforts.

And after I found out error in my argument that I assumed that domain of discourse can be empty. But I found on following reference link that in FOL domain of discourse must be non empty. If we take nonempty condition than my argument breaks down. (:

Reference: https://en.m.wikipedia.org/wiki/First-order_logic

Here the condition of nonempty domain is mentioned.

Please log in or register to answer this question.

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
Welcome to GATE CSE Doubts, where you can ask questions and receive answers from other members of the community.
Top Users Dec 2019
  1. Pratyush Priyam Kuan

    158 Points

  2. Vimal Patel

    118 Points

  3. avistein

    65 Points

  4. srestha

    54 Points

  5. Mk Utkarsh

    49 Points

  6. arya_stark

    46 Points

  7. goxul

    39 Points

  8. Sathuri Bharath

    34 Points

  9. vishal burnwal

    32 Points

  10. Shaik Masthan

    28 Points

Monthly Top User and those within 60% of his/her points will get a share of monthly revenue of GO subject to a minimum payout of Rs. 500. Current monthly budget for Top Users is Rs. 75.
2,317 questions
1,294 answers
6,600 comments
89,719 users