Let $C$ be an equivalence relation on a set $A$. If $A_0 \subset A$, show that the restriction of $C$ to $A_0$ is an equivalence relation
My Attempted Proof
Put $A_0 \subset A$. By definition the restriction of $C$ (where $C \subset A \times A$) to $A_0$ is defined to be $$(C) \ \cap (A_0 \times A_0) = \left\{(x, y) : x \in A_0, y \in A_0, (x,y) \in C\right\}$$
Let $R = C \ \cap (A_0 \times A_0)$
Since $C$ is an equivalence relation on $A$, we have
- $xCx$ (Reflexivity)
- $xCy \implies y C x$ (Symmetry)
- $xCy \ \ \text{and} \ \ yCz \implies xCz$ (Transitivity)
for all $x, y, z \in A$. Since $A_0 \subset A$, all these properties above hold for all $x,y,z \in A_0$ (since $a \in A_0 \implies a \in A$). Thus $R$ is also an equivalence relation on $A_0$
Is my proof correct? If so how rigorous is it? Also any comments on my proof-writing skills, and areas to improve in are greatly appreciated.