A community for students.
Here's the question you clicked on:
 0 viewing
Nodata
 3 years ago
Why \[\forall x \in \emptyset: P(x)\] is equivalent to \[x\in \emptyset \Rightarrow P(x)\]?
Nodata
 3 years ago
Why \[\forall x \in \emptyset: P(x)\] is equivalent to \[x\in \emptyset \Rightarrow P(x)\]?

This Question is Closed

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.3it's just how mathematicians/logicians write things down. it's a matter of notation, and nothing more. they both state the same thing in a different way. they both claim that for any x in the empty set, P(x) is true another trivial example, what if we said: E = the set of all even numbers P(x) = "x + 2 is in E" and we made this claim: For all even numbers e in E, e + 2 is in E. \[\forall e \in E, P(e)\] see how this is an equivalent statement to: "if e is in E, then e + 2 is in E" \[(e \in E) \rightarrow P(e)\]

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.1\[\begin{array}{ccc}\hline\phi&\psi&\phi\Rightarrow\psi\\\hline T&T&T\\T&F&F\\F&T&T\\F&F&T\\\hline\end{array}\]\[\begin{array}{cccc}\hline \forall x \in \emptyset:P(x)&x\in\emptyset &P(x)& x\in\emptyset \Rightarrow P(x)\\\hline T&T&T&T\\\hline\end{array}\]
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.