Anyone know of such definition?

[tgt]

A number is prime if it's only divisors are 1 and itself. We all know that the implication is iff. The convention is to state only the if part. However does anyone know of a mathematical definition whereby it really is defining the if condition only?

i.e a definition like A is true if condition B is satisfied. But A being true does not mean condition B is satisfied?

 

[Hurkyl]

I don't see how such a thing could make sense as a 'definition'.

If

A number is prime if it's only divisors are 1 and itself.​

was a definition, then it would be impossible to conclude anything from "x is prime". (Except for tautologies, of course)

Worse, we could never disprove "n is prime" in any circumstance.

 

[tgt]

I don't see how such a thing could make sense as a 'definition'.

If

A number is prime if it's only divisors are 1 and itself.​

was a definition, then it would be impossible to conclude anything from "x is prime". (Except for tautologies, of course)

Worse, we could never disprove "n is prime" in any circumstance.

That's what I thought when all the textbooks I have read gave definitions with if statements only. But really they mean iff.

However, I wonder if there really is a definition that is if and not iff.

 

[symbolipoint]

"If and Only If", often abbreviated as "iff". Look for information on "biconditional statements". You need two clauses and each must imply the other. This is how you may understand definitions better. This idea should fit very well with your example of prime numbers.

 

[Hurkyl]

However, I wonder if there really is a definition that is if and not iff.

I don't see how such a thing could make sense as a 'definition'. It would be impossible to infer anything from a term 'defined' in such a manner.

 

[HallsofIvy]

A "definition" is (by definition!) saying "this is the same as that" or "this is another name for that". A definition necessarily sets up an equivalence. It must be symmetric and so must be "if and only if".

 

[tgt]

A "definition" is (by definition!) saying "this is the same as that" or "this is another name for that". A definition necessarily sets up an equivalence. It must be symmetric and so must be "if and only if".

There was a senior grad student who said that there could be a term where things imply it but it doesn't imply anything. She must have been wrong?

 

[HallsofIvy]

I have no idea what you mean by "term" here. Are still talking about a definition?

 

[tgt]

I have no idea what you mean by "term" here. Are still talking about a definition?

A term in that context is the name that represents the entity being defined. So prime is a term.

 

[HallsofIvy]

I still don't understand what "there could be a term where things imply it but it doesn't imply anything" means. Statements imply, and are implied by, other statements. "Terms", and "names", don't imply anything!

 

[trambolin]

I think it is not good to use if clauses in definitions exactly for these reasons. My professor used to say "the counting and definitions are the same. You just assign objects from an index set to other objects in other sets."

Use "it is called ... when", "we define ... as the set ..." or whatever your favorite style is but it is better not to overload the "if statement" for the definitions, due to the fact that we become extremely sensitive to such implications after a while, e.g. this thread!