Suppose X ~ N(0,1).
Why can we write P(a≤ X ≤ b) = P(X ≤ b) – P(X ≤ a)
To really answer your question, we need to know how much you understand about continuous distributions.
For example do you understand that for any a, $\displaystyle \mathcal{P}(X=a)=0~?$
From that it follows at once that $\displaystyle \mathcal{P}(X\le a)=\mathcal{P}(X<a).$
So $\displaystyle \mathcal{P}(X\ge a)=1-\mathcal{P}(X<a)=1-\mathcal{P}(X\le a)$.
Thus $\displaystyle =\mathcal{P}(a\le X \le b)=\mathcal{P}(X\le b)-\mathcal{P}(X\le a)$.