Are All Math Definitions 'If and Only If' Statements?

[pivoxa15]

Are all definitions in maths 'if and only if' statement?

One book actaully has A=>B as a definition but I should intepret it as A<=>B as the definition?

 

[quasar987]

Think about it for a second. ;)

For instance, if by "A field is a Galois field if it is of finite cardinality" me meant only that "finite cardinality ==> it's Galois", then it would be without meaning to say that a field is Galois. But our goal is precisely to be able to say "a Galois field" instead of the words "a field of finite cardinality", because it's shorter.

 

[MathematicalPhysicist]

oy vei... (-:

 

[HallsofIvy]

Yes, every definition says "this" and "that" are the same and so is always an "if and only if" statement. Sometimes people get lazy and don't include the "only if" part, but they should! (I've been guilty of that myself, but I'm notoriously lazy! Often I sit at the computer answering silly questions when I should be workin.)

 

[MathematicalPhysicist]

(I've been guilty of that myself, but I'm notoriously lazy! Often I sit at the computer answering silly questions when I should be workin.)

Greg doesn't pay enough for the mentorring, well you can go on a strike like the israeli lecturers it won't get you far though.
(-:

 

[pivoxa15]

For those who voted no, state your reason.