Here is an excerpt from my textbook that I am having trouble understanding:

The cumulative distribution function F of the random variable X is defined for any real number x by
$$F(x) = P\{X\leqslant x \} $$

That is, F(x) is the probability that the random variable X takes on a value that is less than or equal to x. 

All probability questions about X can be answered in terms of its distribution function F

Suppose we want to compute $ P\{a < X \leqslant \ b\} $

This can be accomplished by first noting that the event $\{X \leqslant b\}$ can be expressed as the union of the two mutually exclusive events $\{X \leqslant a \} $ and $ \{a < X \leqslant \ b\} $

I don't see how this could be because $a$ is smaller than $X$ in one, and bigger or equal in the other