Question

b)                 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 the equivalent class of 𝑎, 𝑏 and 𝑐?  

Answer 1

 Let $𝑆 =\{𝑎, 𝑏, 𝑐\}$ and $𝑅 = \{(𝑎, 𝑎), (𝑏, 𝑏), (𝑐, 𝑐), (𝑏, 𝑐), (𝑐, 𝑏)\}$. Let us find the equivalent classes $[𝑎], [𝑏]$ and $[𝑐]$. Taking into account that $[x]=\{s\in S \ |\ (x,s)\in R\}$, we conclude that $[a]=\{a\}, \ [b]=\{b,c\}$ and $[c]=\{b,c\}=[b].$ Therefore, the set of equivalent classes is $\{[a],[b]\}.$