in Mathematical Logic edited by
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24 votes
24 votes

Consider two well-formed formulas in propositional logic

$F_1: P \Rightarrow \neg P$          $F_2: (P \Rightarrow \neg P) \lor ( \neg P \Rightarrow P)$

Which one of the following statements is correct?

  1. $F_1$ is satisfiable, $F_2$ is valid
  2. $F_1$ unsatisfiable, $F_2$ is satisfiable
  3. $F_1$ is unsatisfiable, $F_2$ is valid
  4. $F_1$ and $F_2$ are both satisfiable
in Mathematical Logic edited by
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4 Comments

$F_1: P \Rightarrow \neg P\equiv\neg P\vee \neg P\equiv\neg P$

When we put $P\equiv T$ it will $F$

 and When we put $P\equiv F$ it will $T$

It is called contingency.

Always false called contradiction or unsatisfiable

Always true called valid or tautology

At least one true called satisfiable.

 

$F_2: (P \Rightarrow \neg P) \lor ( \neg P \Rightarrow P)\equiv\neg P\vee P\equiv T$(Always true)

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Correct ME

F1: P => ~p

This statement have logical implication(=>) means  P → ~p is tautology

therefore F1 is always true (therefore F1 is valid statement)
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If this question would be MSQ, then option A) and D), both are correct.
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3 Answers

32 votes
32 votes
Best answer

$F1: P\to \neg P$

    $=\neg P\vee \neg P$

    $=\neg P.$      can be true when P is false ( Atleast one T hence satisfiable)

$F2:  (P\to \neg P)\vee (\neg P\to P)$

     $=\neg P \vee (P\vee P)$

     $=\neg P \vee P$

     $=T.$

VALID

Option (A)

edited by

4 Comments

 

Read the entire comment and answer, you will understand better.

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ok
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Arjun Sir, there's some typo here in F1 

=¬P¬P 

it should be -P v –P 

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2 votes
2 votes

"A valid (true for every set of values)  formula is always satisfiable (true for at least one value)  , but a satisfiable formula may or may not be valid".

as F1 is not valid but satisfiable,

and F2 is valid(which implicitly means is satisfiable too).

therefore OPTION A is the complete solution, but OPTION D is not. 

–2 votes
–2 votes
I can see both are satisfiable i wl go for option b

4 Comments

valid means always true. That is, whatever be the values assigned to the variables, the formula returns true. eg. $p\vee \neg p$
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Then what is your opinion about this which is the most appropriate
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(a) is the most appropriate.
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Answer:

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