Two vectors u,v ∈ V are said to be orthogonal if

[Noxide]

Two vectors u,v ∈ V are said to be orthogonal if <u,v> = 0.


Given the following statement: Two vectors u,v ∈ V are said to be orthogonal if <u,v> = 0.
Is it correct to write it as: if <u,v> = 0, then the two vectors u,v ∈ V are said to be orthogonal

or

Is it correct to write it as: Two vectors u,v ∈ V are said to be orthogonal <=> <u,v> = 0.


Also, can I substitute the word: Provided
for If
in a logical implication?

 

[Fredrik]

Two vectors u,v ∈ V are said to be orthogonal if <u,v> = 0.

"If" should always be interpreted as "if and only if" in a definition. That is a bit weird, but it has become the standard way to write definitions. After defining "orthogonal" this way, you would be correct to say that u and v are orthogonal if and only if <u,v>=0. I don't know why it's standard to write "if" instead of "if and only if" in definitions. Maybe it has something to do with the fact that "<u,v>=0" is a mathematical statement while "Two vectors u,v ∈ V are said to be orthogonal" is a statement about something that people do.

Is it correct to write it as: if <u,v> = 0, then the two vectors u,v ∈ V are said to be orthogonal

or

Is it correct to write it as: Two vectors u,v ∈ V are said to be orthogonal <=> <u,v> = 0.

Logically there's no difference between "if X then Y" and "Y if X", so the first one should be equivalent to the standard definition. However, since it's not in the standard form, it could leave people wondering if you meant something non-standard. I would interpret it as an equivalence, not an implication, because the words "are said to be" are telling me that this is a definition, and definitions are always equivalences. But I would still recommend that you stick to the standard form to minimize confusion.

The second option is a weird mix of English and logic symbols. When you use the equivalence arrow, you should have propositions on both sides of it, not a proposition on the right and the beginning of a definition on the left. You could e.g. write "u,v ∈ V are orthogonal \Leftrightarrow <u,v> = 0". This equivalence is vacuously true, since the proposition on the left is just the proposition on the right written in a different way.

I think "Two vectors u,v ∈ V are said to be orthogonal if and only if <u,v> = 0" would be a good way to write a definition. It makes more sense than the standard way, I think. But if you write one of your definitions this way, you'd have to write all of them this way, unless you'd like to confuse people.

Also, can I substitute the word: Provided
for If
in a logical implication?

I suppose you can, but I wouldn't.

 

[Noxide]

Wow. Thanks so much for the thorough reply!