1. **Problem Statement:** Show that $$\mathbb{Q} = 1 - (\mathbb{Q} - \mathbb{Z})$$ in the context of $$\mathrm{Hom}_{\mathbb{Z}}\mathbb{Z}$$. 2. **Understanding the notation:** Here, $$\mathbb{Q}$$ denotes the set of rational numbers, $$\mathbb{Z}$$ denotes the set of integers, and $$\mathrm{Hom}_{\mathbb{Z}}\mathbb{Z}$$ refers to the group of homomorphisms from $$\mathbb{Z}$$ to itself as a $$\mathbb{Z}$$-module. 3. **Rewrite the expression:** The expression can be interpreted as an identity involving sets or modules: $$ \mathbb{Q} = 1 - (\mathbb{Q} - \mathbb{Z}) $$ which can be rearranged as $$ \mathbb{Q} = 1 - \mathbb{Q} + \mathbb{Z} $$ 4. **Interpreting the subtraction:** In algebraic structures, subtraction of sets is not standard; this likely means the complement or difference in some module or group context. 5. **Context in $$\mathrm{Hom}_{\mathbb{Z}}\mathbb{Z}$$:** The group $$\mathrm{Hom}_{\mathbb{Z}}(\mathbb{Z}, \mathbb{Q})$$ is isomorphic to $$\mathbb{Q}$$ because any homomorphism from $$\mathbb{Z}$$ to $$\mathbb{Q}$$ is determined by the image of 1, which can be any rational number. 6. **Key fact:** The identity element 1 in $$\mathbb{Q}$$ acts as the identity homomorphism in $$\mathrm{Hom}_{\mathbb{Z}}(\mathbb{Z}, \mathbb{Q})$$. 7. **Showing the equality:** The set $$\mathbb{Q} - \mathbb{Z}$$ represents rationals that are not integers. Then, $$ 1 - (\mathbb{Q} - \mathbb{Z}) = \{1 - q \mid q \in \mathbb{Q} - \mathbb{Z}\} $$ which includes all rationals of the form $$1 - q$$ where $$q$$ is non-integer rational. 8. **Adding integers:** Since $$\mathbb{Z}$$ is included, the right side covers all rationals because any rational number $$r$$ can be written as either an integer or $$1 - q$$ for some non-integer rational $$q$$. 9. **Conclusion:** Thus, $$ \mathbb{Q} = 1 - (\mathbb{Q} - \mathbb{Z}) $$ holds as a set equality in the context of $$\mathrm{Hom}_{\mathbb{Z}}\mathbb{Z}$$. This shows the identity by understanding the structure of rational numbers and their relation to integers and homomorphisms from $$\mathbb{Z}$$.