anonymous
  • anonymous
Can someone explain this in plain english, or spanish: \[\forall x \in \emptyset : P(x)\] is TRUE regardless of the value of P(x)
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
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
anonymous
  • anonymous
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.
anonymous
  • anonymous
But that is like a convention because there is no x to make TRUE the statement.

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

anonymous
  • anonymous
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
anonymous
  • anonymous
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
anonymous
  • anonymous
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.
anonymous
  • anonymous
\[x\in \emptyset \Rightarrow P(x)\] totally makes sense to me. What I can't see is how is this equivalent to the universal quantification.
ParthKohli
  • ParthKohli
Sorry, can anyone just tell me what \(\rm P(x)\) stands for?
ParthKohli
  • ParthKohli
I do know that conditional is not equivalent to \(\rm if\cdots then\) statements.
ParthKohli
  • ParthKohli
Oh, I get it.
anonymous
  • anonymous
\[P(x)\] 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 \[x > 0\]
anonymous
  • anonymous
if you give x the value -1 you can say that the proposition is FALSE.
ParthKohli
  • ParthKohli
\[\forall x \in \emptyset :\rm P(x)\]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
  • ParthKohli
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.
ParthKohli
  • ParthKohli
If \(x \in \emptyset\), we know that \(P(x)\) is true so the conditional is true.
ParthKohli
  • ParthKohli
So the statement is always true, and it is given in your original statement that it is true.
ParthKohli
  • ParthKohli
Two statements having the same truth tables are equivalent.
anonymous
  • anonymous
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 \[(x \in \emptyset) \rightarrow P(x)\]
anonymous
  • anonymous
Then can I turn every universal quantification into an implication?
anonymous
  • anonymous
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"
anonymous
  • anonymous
yes you can
anonymous
  • anonymous
generally speaking \[\forall x \in X, P(x)\] is equivalent to \[(x \in X) \rightarrow P(x)\]

Looking for something else?

Not the answer you are looking for? Search for more explanations.