Mathematical Notation - Arrow Sign

Question

What does the $\Rightarrow$ arrow mean when showing working out in maths?

How do we use it appropriately?

Answer 1

The $\Rightarrow$ notation means that if the function on the left hand side of the notation is true, then so is the function on the right hand side of the notation.

So consider $X\Rightarrow Y$. This means that if $X$ is true, then $Y$ is also true.

Answer 2

It stands for "implies that". For example, $x = 2 \implies x^2 = 4$ - if $x$ is $2$, then it is obvious that $x$ squared is $4$; the symbol essentially shows a function here.

Answer 3

The OP use of $\Rightarrow$s is correct. It is the "let" that is syntactically ambiguous. Are you assuming "$a=2^x$" or are you assuming the series of implications? The point is that mathematically trained people can deal with this abuse of notation in most cases and insisting otherwise in a piece of homework would be seen as pedantic. Writing such passages in a research paper is usually frowned upon in the same way as bad grammar would be frowned upon.