Awesome q2a theme
0 votes
24 views

Which of the following statement is valid?

$A)\left \{ \forall \left \{ P\left ( x \right )\vee Q\left ( x \right ) \right \}, \forall x Q\left ( x \right ) \right \}\Rightarrow \exists x P\left ( x \right )$

$B) \left \{ \forall x P\left ( x \right ) , \exists x \sim \left \{ P\left ( x \right ) \wedge Q\left ( x \right )\right \} \right \} \Rightarrow \exists x \sim Q\left ( x \right )$


 We cannot solve it putting simple logic,  We need to use propositional formula here, right?

in Mathematical Logic by (636 points) | 24 views

1 Answer

+1 vote
Best answer

$ \forall \left \{ P\left ( x \right )\vee Q\left ( x \right ) \right \}$  //Either P or Q should be true.

      $ \forall x Q\left ( x \right )$   //Q is always true.

______________________________________

$\therefore  \exists x P\left ( x \right )$ // This means Q(x) is always True so P(x) can be true or false. So it is possible that $ \exists x P\left ( x \right )$ is True but we can't say surely about it.

so $A)\left \{ \forall \left \{ P\left ( x \right )\vee Q\left ( x \right ) \right \}, \forall x Q\left ( x \right ) \right \}\Rightarrow \exists x P\left ( x \right )$ is not valid but it is satisfiable.


 

$ \forall x P\left ( x \right ) $ //P is always true

$ \exists x \sim \left \{ P\left ( x \right ) \wedge Q\left ( x \right )\right \} $ \\ There exist a value of $x$ such that P and Q are not true at same time.

But we know P is always True. hence Q must be false for atleast one value of $x$.

____________________________________

$\therefore  \exists x \sim Q\left ( x \right )$ \\ There exist a value of $x$ for which Q is not true.

so $B) \left \{ \forall x P\left ( x \right ) , \exists x \sim \left \{ P\left ( x \right ) \wedge Q\left ( x \right )\right \} \right \} \Rightarrow \exists x \sim Q\left ( x \right )$ is valid

by (4.1k points)
edited by
0

$\therefore  \exists x \sim Q\left ( x \right ) \equiv \sim \forall x  Q\left ( x \right ) $ 

I couldnot understand, how this line could be correct?

For some $x$ $Q(x)$ doesnot exists, does it mean ,"For all x Q is false or not exists" ?? 

0

It means " there  exist some value of x for which Q is not true"

0

RHS u are using for all

Then how

It means " there  exist some value of x for which Q is not true"

this statement could be true?

0
I thought in reverse.

what would be negation of $\forall x Q(x)$ ? i.e. $\sim \forall x Q(x)$ ?
0
Ok i got it.

LHS

= there  exist some value of x for which Q is not true

i.e. for some value of x Q is true and for some value it is false

i.e. for all values of x Q(x) is not true

=RHS
0
Moreover check this point too

In A) the sign is $\Rightarrow$ and not $\Leftrightarrow$

Then say $\exists x P\left ( x \right )$ could be NULL set too, tha means there exists nothing.

So, why A) is not true??

Take ur time and think this point plz.
0

$\exists P(x)$ means there exist atleast one value of $x$ for which P(x) is true.

so $\exists P(x)$ can't  be null set.

As i already mentioned from the given premises we can't derive the conclusion always. So A is false.

  it is possible that ∃xP(x) is True but we can't say surely about it.

0

Second statement meaning and derivation both are wrong @Satbir

but the result is correct

0
Ok i understood. I can't pull out negation like that. Now corrected please check.
+1
" Q must be false for atleast one case " but not all
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 Jan 2020
  1. shashin

    1168 Points

  2. Vimal Patel

    308 Points

  3. Deepakk Poonia (Dee)

    305 Points

  4. Debapaul

    239 Points

  5. Satbir

    192 Points

  6. SuvasishDutta

    142 Points

  7. pranay562

    132 Points

  8. Pratyush Priyam Kuan

    119 Points

  9. tp21

    112 Points

  10. DukeThunders

    96 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.
3,008 questions
1,516 answers
8,989 comments
89,815 users