Let $R$ be the relation on $\mathbb{Z}$ such that $(x, y) \in \mathbb{Z}$ iff the difference between $x$ and $y$ is a multiple of 10
a) Provide a general proof that $R$ is symmetric b) Determine one equivalence class of $R$
I have problems with part and be of this question.
How do you provide a general proof for the members of a relation?
Are there any good resources on equivalence classes? Whenever I think I understand them I realise I have no clue
For (a), what does it mean that $R$ is symmetric? It means that whenever $(x,y) \in R$, also $(y,x) \in R$. So you could try to prove this by taking an arbitrary element $(x,y)$ of $R$, write down what it means that this element is in $R$, and deduce from that that $(y,x)$ is also in $R$.
Explicitly: Take $(x,y) \in R$. Then $10 \mid x - y$. Therefore also $10 \mid y - x$, so $(y,x) \in R$.
For (b), apparently $R$ is an equivalence relation (not just symmetric, but also reflexive and transitive). What is an equivalence class of $R$? Pick some element of ${\mathbb Z}$, say $x$. The equivalence class $[x]_R$ of $x$ is the set of all $y \in \mathbb Z$ such that $(x,y) \in R$. To describe one equivalence class, just pick a concrete element of $\mathbb Z$, write what its equivalence class is and try to simplify.
Explicitly: The equivalence class $[0]_R$ of $0$ is $[0]_R = \{ y \in \mathbb Z : 10 \mid 0 - y \} = \{ y \in \mathbb : 10 \mid y \}$. So, the equivalence class of $0$ consists of all multiples of $10$.