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Show that $\forall x \ P(x) \lor \forall x \ Q(x) $ and $\forall x \ \forall y \ (P(x) \lor Q(y) )$ are logically equivalent.

I tried to consider three cases:

  • $\forall x \ P(x)$ is true,

  • $\forall x \ Q(x)$ is true or

  • both of them are true. (I'm stuck here and don't have any idea).

Shaun
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S.H.W
  • 4,379

1 Answers1

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Using the method of analytic tableaux: start with the negation of

$$((\forall xP(x)\lor\forall xQ(x)))\leftrightarrow(\forall x\forall y(P(x)\lor Q(y)))\tag{1}$$

then apply a series of contradiction-hunting rules to establish that $(1)$ is what's known as a tautology, like so:

enter image description here

Shaun
  • 44,997
  • I'm not familiar with this method , can you help me please ? – S.H.W Mar 03 '19 at 14:57
  • First, @S.H.W, it assumes the negation of $(1)$. Then it splits into the two possibilities: when the LHS of $\leftrightarrow$ is false while the RHS is true, and v.v. The consequences of each possibility are explored, looking for contradictions. Four paths are made in this example. Can you spot the contradictions? – Shaun Mar 03 '19 at 15:02
  • What about other possibilities ? I mean when LHS and RHS both are true or false . – S.H.W Mar 03 '19 at 15:12
  • Then the iff statement would be true, @S.H.W. – Shaun Mar 03 '19 at 15:14
  • How we can be sure that if both of them are true or false doesn't give us a contradiction ? – S.H.W Mar 03 '19 at 15:18
  • Check the truth table for an iff statement, @S.H.W. – Shaun Mar 03 '19 at 15:21
  • A less glib answer to your follow-up question would be that the method assumes the negation of the iff statement in order to find contradictions, so, by the truth table of an iff, those above are the only possibilities, @S.H.W. – Shaun Mar 03 '19 at 15:43
  • This link solved my problem : https://math.stackexchange.com/questions/10435/negation-of-if-and-only-if Thanks a lot . – S.H.W Mar 03 '19 at 15:59
  • Excellent. I'm glad it's sorted. Don't forget to upvote & accept this answer if it does the job, then, now, @S.H.W. – Shaun Mar 03 '19 at 16:05
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    I accepted the answer , thanks again . – S.H.W Mar 03 '19 at 16:08
  • The diagram above was generated here. – Shaun Oct 30 '22 at 19:20