Does the definition of Φ-invariance require that I(x) = T for all x in S?

[Rasalhague]

Meaning of "we require that"

Wikipedia: Dynamical system:

In particular, for S to be Φ-invariant, we require that I(x) = T for all x in S. That is, the flow through x should be defined for all time for every element of S.

Does this mean

(1) If I(x) = T for all x in S, then S is Φ-invariant?

(2) If S is Φ-invariant, then I(x) = T for all x in S?

(3) If and only if S is Φ-invariant, then I(x) = T for all x in S?
(I.e. The above definition of Φ-invariance of S is equivalent to the the requirement that I(x) = T for all x in S.)

 

[torquil]

The property "I(x) = T for all x in S" is a consequence of the fundamental requirement for S to be phi-invariant, i.e. that phi(t,x) is in S, for all x in S and all t in T. They only state it there because it might not be immediately obvious for all.

The statement simply says that "I(x) = T for all x in S" is a property of S, if S is phi-invariant. It does not say that it is the only property. The fundamental definition, stated above the quoted sentence, contains this property as a special case.

So your statement (1) is incorrect.

Your statement (2) is correct, but only because it is true by definition. It's like saying "If G is a group, then G contains an identity". Of course it does, since a group has an identity by definition...

Your statement (3) is incorrect. "I(x) = T for all x in S" is not the only requirement in the definition of phi-invariance. The full requirement is stated on the Wikipedia page above the sentence you quoted.

I recommend that you get an intuitive understanding of what the flow phi(t,x) represents and what it means for S to be phi-invariant. Then all this will be obvious to you

 

[disregardthat]

"For B to be true, we require A" means that B implies A.

 

[Mark44]

"For B to be true, we require A" means that B implies A.

I think you have this backwards. The above means that A implies B.

 

[disregardthat]

I think you have this backwards. The above means that A implies B.

Does "for S to be a set of four elements, we require that S is non-empty" mean ("S is non-empty" implies "S is a set of four elements")?

 

[Mark44]

No. S being non-empty is a necessary but not sufficient condition for S to be a set of four elements.

 

[disregardthat]

No. S being non-empty is a necessary but not sufficient condition for S to be a set of four elements.

That A is required doesn't mean that A is sufficient, it means it is necessary. Torquil got it right. Or perhaps there is some sort of convention that I am unaware of, but this is at least not ordinary usage of the word 'require'.

EDIT: Wiktionary defines:

Requirement: A necessity or prerequisite; something required or obligatory.

 

[Rasalhague]

Thanks Torquil, and everyone else who's replied. I guessed (2), as the literal meaning, but I wasn't sure if it was part of a definition and therefore subject to the rule "if = iff in definitions".

I recommend that you get an intuitive understanding of what the flow phi(t,x) represents and what it means for S to be phi-invariant. Then all this will be obvious to you

Good plan! Actually I think I see why now. If S is Φ-invariant, then I(x) = T for all x in S because of the closure property of a monoid. But I(x) = T for all x in S doesn't guarrantee that there doesn't exist an x in S and a t in T such that phi(x,t) is not in S.

 

[Rasalhague]

"For B to be true, we require A" means that B implies A.

Thanks, Jarle. That makes sense.

 

[Landau]

Yes, that makes sense. However, sometimes the use of 'we require that...' is used while defining something. You know how one says things like 'f is continuous if the inverse image of every open set is open' while actually 'iff' is meant instead of 'if'... You could also say something like 'for f to be continuous, we require that every open set has open inverse image', although this is perhaps not too common.

 

[Rasalhague]

Yes, that makes sense. However, sometimes the use of 'we require that...' is used while defining something. You know how one says things like 'f is continuous if the inverse image of every open set is open' while actually 'iff' is meant instead of 'if'... You could also say something like 'for f to be continuous, we require that every open set has open inverse image', although this is perhaps not too common.

As I mentioned in #8, this was one reason for my uncertainty.

 

[Landau]

Wow, I completely missed your first line in post #8. Sorry.

 

[Rasalhague]

Wow, I completely missed your first line in post #8. Sorry.

No problem! It was still useful to get confirmation that that is indeed a possible source of ambiguity in the context of a definition.