Convex piecewise linear functions - partitioned vs max representation

Question

I know that convex piecewise linear function $f$ can be defined in two closely related ways -

1. There exist $Q \subseteq P(X)$ s.t. $\bigcup Q = X$ and on every $A \in Q$ the function is linear (Example).

2. There exist a set $\{(a_1,b_1),\ldots,(a_m,b_m)\}$ s.t. $f(x) = \max_{i\in[m]} a_i^T x + b_i$ (Example).

It makes sense that if $Q$ is a partition of $X$ it should be that $|Q| = m$ but I can't seem to prove it. Is my guess correct?