Sufficient/necessary vs. weaker stronger I know what sufficient resp. necessary means, but I'm confused, when our professor uses the terminology weaker resp. stronger. I couldn't yet find out, what the translation of these pair of words to the pair necessary stronger is (at least I'm guessing that A being weaker/stronger than B is just a different way to say that A is sufficient/necessary for B) ?
 A: I can think of two relevant ways in which weaker and stronger are used. A statement $A$ is stronger than a statement $B$ if $A$ implies $B$; $A$ is weaker than $B$ if $B$ implies $A$. Equivalently, $A$ is stronger than $B$ if $A$ is sufficient for $B$, and $A$ is weaker than $B$ if $A$ is necessary for $B$.
We also speak of one theorem being stronger or weaker than another. Consider the theorems $A\Rightarrow B$ and $A'\Rightarrow B\,'$. If the hypothesis $A$ is weaker than the hypothesis $A'$, or the conclusion $B$ is stronger than the conclusion $B\,'$ (or both), the theorem $A\Rightarrow B$ is stronger than the theorem $A'\Rightarrow B\,'$ (and of course $A'\Rightarrow B\,'$ is weaker than $A\Rightarrow B$).
Here’s an example of this usage. Consider the following two theorems.

Theorem 1. Let $X$ be a complete metric space, and suppose that $\{G_n:n\in\Bbb N\}$ is a family of dense open subsets of $X$; then $\bigcap_{n\in\Bbb N}G_n\ne\varnothing$.
Theorem 2. Let $X$ be a complete metric space, and suppose that $\{G_n:n\in\Bbb N\}$ is a family of dense open subsets of $X$; then $\bigcap_{n\in\Bbb N}G_n$ is dense in $X$.

Since a dense subset of $X$ must be non-empty, the conclusion of Theorem 2 implies the conclusion of Theorem 1: the conclusion $\bigcap_{n\in\Bbb N}G_n$ is dense in $X$ is stronger than the conclusion $\bigcap_{n\in\Bbb N}G_n\ne\varnothing$. This means that Theorem 2 is stronger than Theorem 1: it reaches a stronger conclusion from the same hypothesis. In a sense it says more.
Here we strengthened Theorem 1 by strengthening its conclusion. One can also strengthen a result by using a weaker hypothesis to reach the same conclusion; here again the outcome is a theorem that in a sense says more, because it uses less information to reach the same conclusion.
