Definitions and if and only if statements

[lonewolf5999]

I'm trying to learn some analysis on my own, and as this is the first proof-based book I'm reading, I have a basic question about definitions I was hoping someone could help me with. For example, the book I'm reading says that: Given a subset of the real numbers A, b is an upper bound of A if every element a of A is less than or equal to b. I rephrase this definition of the upper bound as: if every element a of A is less than or equal to b, then b is an upper bound of A.

My question is: is this definition an if and only if statement? That is, is the statement "If b is an upper bound of A, then every element a of A is less than or equal to b" also true? It seems like it should be, but I was hoping to get some confirmation or clarification on this. More generally, is it safe to assume that all definitions are if and only if statements? If not, is there any way to tell when they aren't?

 

[micromass]

Hi lonewolf5999!

Yes, it is safe to assume that every definition is an "if and only if"-statement.

 

[disregardthat]

A definition is a tautologically true predicate. It is essentially a substitution of terms.

Is "u = 2" an iff statement, where u is previously undefined? Well, the "if and only if" would consist of "x = 2 <-> u = x". The iff-part is uneccesary, and adds nothing to what has already been defined. In formal systems it might be otherwise. Definitions might (or might not) be stated in terms of an iff-statement. But that is not necessarily directly relevant.

 

[TylerH]

As they've already said: Yes, definitions are defining what is logically equivalent.

But, I want to mention something else. Usually, one should only name things you intend to mention again. For example: "if every element a of A is less than or equal to b, then b is an upper bound of A." The "a" is unnecessary. The meaning of "if every element of A is less than or equal to b, then b is an upper bound of A" is exactly the same and doesn't leave the reader wondering "wtf is the a for"? Now, if you rephrased it "If, for all a in A, a is less than or equal to b, then b is the upper bound of A," then it is clear the naming of a is necessary.

Not to nitpick. I'm just now learning to write proofs and logical statements myself. :)