Consider the equivalence relation on Z ! Z given by (m, n)R(p, q) if and only if mq = np:
(a) Find the equivalence class represented by (2, 5).
(b) Describe the set S of the equivalence classes determined by R.
Until now I didn't get the idea of equivalence classes, how can I solve the above.
Note: It's NOT a homework, I'm just doing some revision and problem solving.
(a) The equivalence class represented by $(2,5)$ are all the ordered pairs $(x, y)\in Z$ such that $(2,5)R(x,y)$:
$$(2,5)R(x,y) \Longleftrightarrow 2y = 5x \Longleftrightarrow \frac{x}{y} = \frac{2}{5}$$
Any ordered pair $(x,y)$ that satisfies this equation is thus part of the equivalence class of $(2,5)$.
(b) The set $S$ of the equivalences classes of $R$ in $Z$, is given by:
$$S = \{\left [ (a,b) \right ]_R : (a,b) \in Z \}$$
where $[\cdot]_R$ means the equivalence class of $\cdot$ relatively to $R$. That set is a partition of $Z$, since all equivalences classes are disjoint $$ (a,b)R(x,y) \Longrightarrow \left [ (a,b) \right ]_R = \left [ (x,y) \right ]_R $$ and the union of all equivalent classes is $Z$ (the domain of $R$ is all ordered pairs of $Z$).