So, I can see the difference between something like:

A. A car is green if it is made in England.


B. A car is green if and only if it is made in England.

Then, if you had a Russian-made green car, it would be true for A. but not for B. So B is a stricter form of A. I'm trying to see how I can apply this logic to the statement

A function $f: A \to B$ is surjective if and only if for all $b \in B$, there exists an $a \in A$ such that $f(a) = b$.

I think what this means that if it were just normally implied (if x then y), you could have a surjection without the property: $\forall b \in B, \exists a \in A: f(a) = b$; i.e. there could be another property that allows a function to be surjective. But in saying if and only if, we are ensuring that a function can only be surjective if it has this property?


You're right.

You may find, however, that people take this distinction less literally in definitions than in theorems. That is, if a theorem says "If a function from $\mathbb R$ to $\mathbb R$ is continuous, it takes on all values between any two of its function values", it really means only that and leaves open the possibility that the "only if" statement doesn't hold; however, in definitions (and what you're quoting is the definition of "surjective"), "if" is often used to mean "if and only if"; that is, you may find definitions like "a number is said to be even if it is divisible by $2$", intended to mean "a number is said to be even if and only if it is divisible by $2$".

  • $\begingroup$ Thankyou, this is exactly what I was worried about. I was looking at statements in my book that were just implications, but I could not find examples where the stronger biconditional implication didn't hold. I guess for most practical uses it doesn't matter, but pedagogically it's a little confusing :) $\endgroup$ – njp Dec 2 '12 at 16:39

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