Every arbitrary equivalence relation R in a set X divides X into mutually disjoint subsets (A i ) called partitions or subdivisions of X satisfying the following conditions:
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All elements of A i are related to each other for all i
Sir , I can't under stand the line means. Plz explain in brief.
Hi!
Here is the answer to your query.
The statement given by you can be verified by the following example.
Consider equivalence relationis R = {(a, b): a = b or a = −b for a, b ∈ R}.
[Proved in study material]
Equivalence class of 0 = [0] = {0} = A1 (say) [By the definition of give relation R]
Equivalence class of 1 = [1] = {1, − 1} = A2 (say)
Equivalence class of 2 = [2] = {2, − 2} = A3 (say)
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So, A1, A2, A3, A4 …. are partitions of R.
Thus, the relation R in the set R divides R into mutually disjoint subsets (Ai) called partitions or subdivisions of R.
Now, you an also see that all elements of Ai are related to each other for all i.
This can be verified by taking a example.
Take the class of [1] = {1, − 1} = A2
Here, 1 R 1 [As 1 = 1]
1 R –1 [As 1 = – (–1)]
Cheers!