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Nodata
 3 years 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)
Nodata
 3 years 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)

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binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.2this 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

Nodata
 3 years ago
Best ResponseYou've already chosen the best response.0I'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.

Nodata
 3 years ago
Best ResponseYou've already chosen the best response.0But that is like a convention because there is no x to make TRUE the statement.

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.2doesn'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

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.2in other words, the only thing that causes a conditional statement to be false is if the conditional is true AND the consequent is false

Nodata
 3 years ago
Best ResponseYou've already chosen the best response.0I 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.

Nodata
 3 years ago
Best 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.

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

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

Nodata
 3 years ago
Best 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 \[x > 0\]

Nodata
 3 years ago
Best ResponseYou've already chosen the best response.0if you give x the value 1 you can say that the proposition is FALSE.

ParthKohli
 3 years ago
Best 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.

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.0So \(x \in \emptyset \implies P(x)\) is true. If \(x \not \in \emptyset\), then the first part of the conditional is false. So the statement is true.

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.0If \(x \in \emptyset\), we know that \(P(x)\) is true so the conditional is true.

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.0So the statement is always true, and it is given in your original statement that it is true.

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.0Two statements having the same truth tables are equivalent.

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.2what 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 \[(x \in \emptyset) \rightarrow P(x)\]

Nodata
 3 years ago
Best ResponseYou've already chosen the best response.0Then can I turn every universal quantification into an implication?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.2the 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"

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