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\\}."
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