Question

\[a\implies b\] does it mean \[b\implies a\] ?

Answer 1

yes

Answer 2

But \[¬B \implies ¬A\]
is not neccessarily true.

Answer 3

To illustrate the previous sentence:
I'm a male => I'm human is true
I'm human => I'm a male is false
I'm not human => I'm not a male is true

Answer 4

a<=>b
does it mean
b<=>a?

Answer 5

a⟹b is not logically equivalent to b⟹a
however a⇔b is logically equivalent to b⇔a

Answer 6

I often find useful to think with diagrams:

if a then b (if I'm in A, then I'm in B). So it's clear the opposite is not necessary true.
But if I'm not in B, then I'm not in A