What is the difference between $\forall$ and $\implies$

Question

Hellow, what is the difference? between $$\forall{a}\in{A}:\ldots$$ and $${a}\in{A}\Rightarrow{}\ldots$$

Answer 1

We can write:$$\forall a\in A [P(a)]$$This in order to state that every element of set $A$ has property $P$.

Another way of expressing this is:$$\forall a[a\in A\implies P(a)]$$ In words: for every $a$ it is true that it has property $P$ if $a$ happens to be an element $A$.

Sometimes we just leave out the quantor and abbreviate this as:$$a\in A\implies P(a)$$This with in the back of our mind the knowledge that we think of all $a$.

I hope this makes things more clear for you.