Is (∀v Fv -> p) Equivalent to (∃u Fu -> p)?

In summary, the conversation is about the equivalence between two statements involving quantifiers and a proposition. The conditions for this equivalence are that the variable v occurs free in Fv at all and only in the places where u occurs free in Fu, and that the proposition p contains no free occurrences of v. The proof of this equivalence can be shown by representing the implication as a disjunction and using de Morgan's law for quantifiers.
  • #1
agapito
49
0
Consider the equivalence:

(∀v Fv -> p) <=> (∃u Fu -> p)

Where variable v occurs free in Fv at all and only those places that u occurs free in Fu, and p is a proposition containing no free occurences of variable v.

Can someone please offer a proof of such equivalence. Many thanks. am
 
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  • #2
I am not sure what it means to occur free "at all", and I don't understand the phrase "only those places that u occurs free in Fu". What is claimed about the places where u occurs free?

Perhaps your equivalence is $\forall v\,(Fv\to p)\iff (\exists u\,Fu)\to p$ or $(\forall v\,Fv)\to p\iff \exists u\,(Fu\to p)$. This is easy to show if you represent $A\to B$ as $\neg A\lor B$ and use de Morgan's law for quantifiers.
 

FAQ: Is (∀v Fv -> p) Equivalent to (∃u Fu -> p)?

What is proof of equivalence?

Proof of equivalence is a mathematical or logical method used to show that two systems, concepts, or theories are equivalent. It involves demonstrating that the properties, behaviors, or outcomes of the two systems are identical or interchangeable.

Why is proof of equivalence important?

Proof of equivalence is important because it allows us to establish a relationship between two seemingly different systems or concepts. It helps us to understand the connections and similarities between them, and can also be used to simplify complex problems by reducing them to a known equivalent system.

What are some common techniques used for proof of equivalence?

Some common techniques used for proof of equivalence include mathematical induction, direct proof, contradiction, and structural induction. These techniques involve breaking down the problem into smaller, more manageable parts and showing that each part is equivalent to the other.

Can proof of equivalence be applied to any type of system or concept?

Yes, proof of equivalence can be applied to any type of system or concept as long as there is a clear definition of what it means for the two systems to be equivalent. It is a versatile tool that can be used in various fields such as mathematics, computer science, and physics.

How can I improve my skills in providing proof of equivalence?

Improving your skills in providing proof of equivalence requires practice and a good understanding of mathematical and logical concepts. You can also benefit from studying and analyzing existing proofs of equivalence, as well as seeking guidance from experienced mathematicians or scientists.

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