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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
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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binarymimicBest ResponseYou've already chosen the best response.2
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

NodataBest ResponseYou've already chosen the best response.0
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

NodataBest ResponseYou've already chosen the best response.0
But that is like a convention because there is no x to make TRUE the statement.
 one year ago

binarymimicBest ResponseYou've already chosen the best response.2
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

binarymimicBest ResponseYou've already chosen the best response.2
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

NodataBest ResponseYou've already chosen the best response.0
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

NodataBest ResponseYou've already chosen the best response.0
\[x\in \emptyset \Rightarrow P(x)\] totally makes sense to me. What I can't see is how is this equivalent to the universal quantification.
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.0
Sorry, can anyone just tell me what \(\rm P(x)\) stands for?
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.0
I do know that conditional is not equivalent to \(\rm if\cdots then\) statements.
 one year ago

NodataBest ResponseYou've already chosen the best response.0
\[P(x)\] 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

NodataBest ResponseYou've already chosen the best response.0
if you give x the value 1 you can say that the proposition is FALSE.
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.0
\[\forall x \in \emptyset :\rm P(x)\]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

ParthKohliBest ResponseYou've already chosen the best response.0
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

ParthKohliBest ResponseYou've already chosen the best response.0
If \(x \in \emptyset\), we know that \(P(x)\) is true so the conditional is true.
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.0
So the statement is always true, and it is given in your original statement that it is true.
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.0
Two statements having the same truth tables are equivalent.
 one year ago

binarymimicBest ResponseYou've already chosen the best response.2
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

NodataBest ResponseYou've already chosen the best response.0
Then can I turn every universal quantification into an implication?
 one year ago

binarymimicBest ResponseYou've already chosen the best response.2
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

binarymimicBest ResponseYou've already chosen the best response.2
generally speaking \[\forall x \in X, P(x)\] is equivalent to \[(x \in X) \rightarrow P(x)\]
 one year ago
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