I was trying to understand the meaning or the definition of what a "stronger statement" is formally. I came across the following definition (from Mattuck's Analysis book):
Stronger and weaker. If $A \implies B$ is true, but $B \implies A$ is false, we say: A is a stronger statement than B; B is weaker than A.
and I was wondering, why is the definition that way? Is there a conceptual/intuitive way to explain this? I know this is just "the definition" but I was trying to understand why it is that way.
In fact I create some type of "memory device" (not sure what else to call it) to remember/justify it to myself. I draw the following diagram:
and then notice that whenever x is in A it means it must be is B also. Therefore, being in A implies being in B. Furthermore, if x is in B it doesn't necessarily always be in A, so the converse is not always true automatically. The only issue I have with my memory device is that I obviously just re-define what the statement A means to be a very specific set membership statement. So I assume its a fine memory device but its oversimplifing things "cheating" in some way. i.e. its not a proof nor I expect it to be the "real reason" why the definition holds.
So I was looking to understand a better way to understand what a stronger statement means without "cheating".
As pointed out by the comment, A is stronger because it says everything B says and more. I guess for me intuitively that would have meant that A is a bigger set, but in my memory device that translated to a smaller set, which makes it confusing to me. Anyone know why?