# Is implication true if two statements are always the case?

I have a task that requires me to show that under a certain set of circumstances, a set has property A if and only if it has property B.

I can show that under the given circumstances, the set always has property A. I can also show that it always has property B.

Since these properties are always the case, is seems that it is true that $$A \Leftrightarrow B.$$

My question is: Is this a valid proof of an IFF statement?

I stress I prove no connection or relationship of implication between A and B, only that both are always true.

Edit. Here is, more concretely, the specific task: Given two metric spaces, $$(X,d_1)$$ and $$(X,d_2)$$, where $$d_1$$ and $$d_2$$ are two equivalent metrics, show that $$d_1$$ is complete if $$d_2$$ only if the other is complete.

Edit 2. I need to show that $$(X,d_1)$$ is complete iff $$(X,d_2)$$ is complete. I know three things:

1. $$kd_1(x,y) \leq d_2(x,y) \leq Kd_1(x,y)$$. (X,d_1)$$and$$(X,d_2).
2. A sequence that is Cauchy in $$d_1$$ is convergent in $$d_2$$.
3. Any sequence that converges in $$d_1$$ converges in $$d_2$$ and vice versa.

So, take any Cauchy sequence in $$d_1$$. By 2) it is convergent in $$d_2$$. By 1), it is convergent in $$d_1$$. Hence, every Cauchy sequence in $$d_1$$ is convergent in $$d_1$$. Hence, $$(X,d_1)$$ is complete. In the same way, it can be shown that $$(X,d_2)$$ is complete.

Since both are complete independently of one another, the "IFF" statement holds.

Does this argument hold?

• In the setting of classical logic, yes, that is a proof of the equivalence.
– Ian
Commented Jul 14 at 15:34
• This seems true but not noteworthy Commented Jul 14 at 15:35
• $A \land B \Rightarrow (A \Leftrightarrow B)$ is true even constructively... Commented Jul 14 at 15:51
• You would have to give the actual wording to ensure that you were not misreading that. I can imagine something like this too : let the circumstances be $C$ , then we may want $(C \land A) \iff (C \land B)$ or $(C \implies A) \iff (C \implies B)$ or $(C \iff A) \iff (C \iff B)$ !
– Prem
Commented Jul 14 at 15:59
• I strongly suspect you have misread something. The statement labeled 1 is (probably) simply the definition of equivalent metrics (I say “probably” because there are other definitions of “equivalent” metrics but I don’t understand the problem at all if it’s not this definition). The other two numbered statements are not necessary as premises of the theorem. Statement 2 in particular should not need to be true in all cases, so it’s not something you would naturally “know.” Commented Jul 14 at 18:51

I think some people misunderstand that if a statement $$C$$ implies both $$B$$ and $$A$$, then in general $$A \Leftrightarrow B$$ does NOT hold. Therefore, in my opinion, you need to specify your question a little more. For example, if the set $$A \subset \mathbb{R}$$ is finite for given circumstances. it always has a minimum and a maximum. In general, however, it does not follow that a set has a minimum iff it has a maximum.

• Thanks, I added the full details. Admit I'm confused.. @Noctis Commented Jul 14 at 18:01
• Your argument that you have written down is correct. Of course, it can be described in more detail, but you got the core right and understood it. Commented Jul 14 at 18:38

There are multiple ways you can prove a statement such as this:

The fact that $$d1,d2$$ are equivalent metrics won't play much of a role in the structure of your proof, consider this schema:

Your universal assumption, this is true regardless: Assume $$d1,d2$$ are equivalent

1. Assume $$A$$, show $$B$$ and Assume $$B$$, show $$A$$ (So-called direct proof)

2. Show ($$A$$ and $$B$$) OR ($$\lnot B$$ and $$\lnot A$$) this technique is often avoided since it leads to unnecessary complications which you will see now:

In order to show $$P\lor Q$$, in this case, our universal assumption will hold, now negate one of the disjuncts, and show the other must be true.

As you can see, universal assumptions don't play much of a role in your logical structure, but they are important in that you might need to use the universal assumption to flesh out your argument.