Question

If $R$ is an arbitrary equivalence relation in an arbitrary set $X$, then $R$ divides $X$ into mutually disjoint subsets $A_i$ called partitions or subdivisions of $X$ satisfying

• A. all elements of $A_i$ are related to each other, for all $i$
• B. no element of $A_i$ is related to any element of $A_j,i\ne j$
• C. $\cup A_j=X$ and $A_i\cap A_j=\varphi ,i\ne j$
• D. All of the above

Answer 1

Answer: (D)

Given, an arbitrary equivalence relation $R$ in an arbitrary set $X,R$ divides $X$ into mutually disjoint subsets $A_i$ called partitions or subdivisions of $X$.
Then Ai's satisfying
(i) all elements of $A$ are related to each other, for all $i$.
(ii) no element of $A$ is related to any element of $A_j,i\ne j$.
(iii) $\cup A_j=X$ and $A\cap A_j=\varphi ,i\ne j$.
The subsets $A$ are called equivalence classes.