Question

Let 𝑆 = {𝑎, 𝑏, 𝑐} and 𝑅 = {(𝑎, 𝑎), (𝑏, 𝑏), (𝑐, 𝑐), (𝑏, 𝑐), (𝑐, 𝑏)}, find [𝑎], [𝑏] and [𝑐] (that is the equivalent class of a, b, and c). Hence or otherwise  find the set of equivalent class of 𝑎, 𝑏 and 𝑐?

Answer 1

Let $𝑆 = \{𝑎, 𝑏, 𝑐\}$ and $𝑅 = \{(𝑎, 𝑎), (𝑏, 𝑏), (𝑐, 𝑐), (𝑏, 𝑐), (𝑐, 𝑏)\}$. Taking into account that $[x]=\{y\in S:(y,x)\in R\}$, we conclude that  $[𝑎]=\{a\},\ [b]=\{b,c\},\ [c]=\{ c, b\}=[b].$ The set of equivalent classes is $\{[a],[b]\}.$