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# Examples of Equivalence Relations

TOPO-JPEVIB

Which of the following are equivalence relations on set $X$?

Select ALL that apply.

A

$X$ is the set of finite subsets in $\mathbb{R}^4$, $\forall a,b\in X$, $a\sim b$ iff $span_\mathbb{Q} a=span_\mathbb{Q} b$.

B

$X$ is the set of infinite sequences of real numbers. $\forall \{a_n\}, \{b_n\}\in X$, $\{a_n\}\sim\{b_n\}$ iff $\lim_{n\rightarrow\infty}(a_n-b_n)=0$.

C

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

D

$X=\mathbb{R}$, $\forall x,y\in X$, $x\sim y$ iff $x-y\not\in\mathbb{Q}$.

E

$X=\mathbb{Q}$, $\forall x,y\in X$, $x\sim y$ iff $x-y\in\mathbb{Q}$.