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No-data Group Title

Can someone explain this in plain english, or spanish: \[\forall x \in \emptyset : P(x)\] is TRUE regardless of the value of P(x)

  • one year ago
  • one year ago

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  1. binarymimic Group Title
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    this is vacuously true for all elements, x, in the empty set, P(x) is true there is no x to falsify the claim of P(x) so you can conclude the statement is true

    • one year ago
  2. No-data Group Title
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    I've been doing some research and I found that this is a shorthand for \[x\in \emptyset \Rightarrow P(x) \] Which is TRUE, but I don't see the relationship.

    • one year ago
  3. No-data Group Title
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    But that is like a convention because there is no x to make TRUE the statement.

    • one year ago
  4. binarymimic Group Title
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    doesn't matter the only thing that causes a condition to be false is if "True implies False" in this case, the conditional statement is always false, so it can imply anything, and the entire statement will always evaluate to true

    • one year ago
  5. binarymimic Group Title
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    in other words, the only thing that causes a conditional statement to be false is if the conditional is true AND the consequent is false

    • one year ago
  6. No-data Group Title
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    I understand what you say about the conditional statement, but what I'm not sure is how you can go from the universal quantifier to a conditional statement.

    • one year ago
  7. No-data Group Title
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    \[x\in \emptyset \Rightarrow P(x)\] totally makes sense to me. What I can't see is how is this equivalent to the universal quantification.

    • one year ago
  8. ParthKohli Group Title
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    Sorry, can anyone just tell me what \(\rm P(x)\) stands for?

    • one year ago
  9. ParthKohli Group Title
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    I do know that conditional is not equivalent to \(\rm if\cdots then\) statements.

    • one year ago
  10. ParthKohli Group Title
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    Oh, I get it.

    • one year ago
  11. No-data Group Title
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    \[P(x)\] is a predicate with a parameter x. Once you know the value of the parameter x you can say it becomes a proposition and you can say if its TRUE or FALSE. For example \[x > 0\]

    • one year ago
  12. No-data Group Title
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    if you give x the value -1 you can say that the proposition is FALSE.

    • one year ago
  13. ParthKohli Group Title
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    \[\forall x \in \emptyset :\rm P(x)\]This is true, means that \(\rm P(x)\) is true for all \(x\) in empty set. If a conditional is true, then: * The first value is false. * If not false, then the second value must be true. In other words \(1\implies 0\) is false.

    • one year ago
  14. ParthKohli Group Title
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    So \(x \in \emptyset \implies P(x)\) is true. If \(x \not \in \emptyset\), then the first part of the conditional is false. So the statement is true.

    • one year ago
  15. ParthKohli Group Title
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    If \(x \in \emptyset\), we know that \(P(x)\) is true so the conditional is true.

    • one year ago
  16. ParthKohli Group Title
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    So the statement is always true, and it is given in your original statement that it is true.

    • one year ago
  17. ParthKohli Group Title
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    Two statements having the same truth tables are equivalent.

    • one year ago
  18. binarymimic Group Title
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    what i was trying to say earlier was that a universal quantifier states a case for any x in the empty set so an equivalent statement is exactly as you have written you can state the case for any x in the empty set as \[(x \in \emptyset) \rightarrow P(x)\]

    • one year ago
  19. No-data Group Title
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    Then can I turn every universal quantification into an implication?

    • one year ago
  20. binarymimic Group Title
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    the only difference is in the notation. \[\forall x \in \emptyset, P(x)\] just says "for any x in the empty set, P(x) is true" whereas \[(x \in \emptyset) \rightarrow P(x)\] just says "if x is in the empty set, then P(x) is true"

    • one year ago
  21. binarymimic Group Title
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    yes you can

    • one year ago
  22. binarymimic Group Title
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    generally speaking \[\forall x \in X, P(x)\] is equivalent to \[(x \in X) \rightarrow P(x)\]

    • one year ago
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