The tower of path algebras associated to a tower of finite dimensional $C^*$-algebras is isomorphic to the original tower

Let $ A_0\subseteq A_1\subseteq…$ be an infinite tower of unital inclusions of finite dimensional $ C^*$ -algebras and $ B_0\subseteq B_1\subseteq …$ be its associated infinite tower of path algebras. Then how to show that these two towers are tower isomorphic ?

Comparing the definitions of $K$-theory and $K$-homology for $C*$-algebras

In Higson and Roe’s Analytic K-homology, for a unital $ C*$ -algebra $ A$ , the definitions of K-theory and K-homology have quite a similar flavor.

Roughly, the group $ K_0(A)$ is given by the Grothendieck group of homotopy classes of matrix algebra projections over $ A$ .

On the other hand, $ K^0(A)$ is the Grothendieck group of homotopy classes of even Fredholm modules over $ A$ (or more correctly unitary classes of Fredholm modules).

Now in contrast to the $ K_0(A)$ case, every element of $ K^0(A)$ contains a representative Fredholm module. (For the inverse of the class of $ (H,\rho,F)$ , just take the class of $ (H^{\text{op}},\rho,-F)$ , where op denotes the opposite grading.

Does this means that taking the Grothendieck is not strictly necessarily to define $ K^0(A)$ ? Could one just as well define it as the monoid of homotopy classes of Fredholm modules, and then prove that it was a group?

If this is true, then is there any deeper philosophical reason why $ K_0$ requires us to introduce inverses, while $ K^0$ does not?