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Equivalence Classes Generated by a Relation

TOPO-ZXB1WG

The equivalence relation generated by a relation $R$ is the smallest equivalence relation containing $R$.

Which of the following statements are TRUE?

Select ALL that apply.

A

If $R\subseteq R'$, the equivalence relation generated by $R$ can not have less equivalence classes than the equivalence relation generated by $R'$.

B

$X$ is the set of subsets of $\mathbb{Z}$, $\forall A,B\in\mathbb{Z}$, $A\sim B$ iff there is bijection from $A$ to $B$ and a bijection from $\mathbb{Z}-A$ to $\mathbb{Z}-B$. Then there is a bijection from $X/\sim$ to $\mathbb{Z}$.

C

Let $X$ be the set of polynomials on $\mathbb{R}$ with degree greater than 2, $\sim$ is the equivalence relation generated by the divisible relation. Then there are infinitely many equivalence classes of $\sim$.

D

$X=\{2,3,\dots 20\}$, $\sim$ is the equivalence relation generated by the divisible relation. $X/\sim$ has less than 5 elements.