Question

List the members of the equivalence relation on (1, 2, 3, 4) defined by the following partition. Find the equivalence classes [1], [2], [3] and [4]


1. {(1, 2), (3, 4))


2. {{1, 2, 3), (4))


3. {{1}, {2}, {3}, {4}}

Answer 1

Let us list the members of the equivalence relation $R$ on $X=\{1, 2, 3, 4\}$ defined by the following partition $P$ . Find the equivalence classes $[1], [2], [3]$ and $[4].$

It is well-known that $(a,b)\in R$ if and only if $a,b\in M$ for some $M\in P.$ The equivalence class generated by $a$ is $[a]=\{x\in X:(a,x)\in R\}.$

1. $\{\{1, 2\}, \{3, 4\}\}$

It follows that $R=\{(1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(4,3),(4,4)\}.$

$[1]=\{1,2\}=[2],\ \ [3]=\{3,4\}=[4].$

2. $\{\{1, 2, 3\}, \{4\}\}$

It follows that $R=\{(1,1),(1,2),(2,1),(2,2),(3,3),(1,3),(3,1),(2,3),(3,2),(4,4)\}.$

$[1]=\{1,2,3\}=[2]=[3],\ \ [4]=\{4\}.$

3. $\{\{1\}, \{2\}, \{3\}, \{4\}\}$

It follows that $R=\{(1,1),(2,2),(3,3),(4,4)\}.$

$[1]=\{1\},\ \ [2]=\{2\},\ \ [3]=\{3\},\ \ [4]=\{4\}.$