Equivalence Relations, Partitions and Equivalence Classes

Question

Consider the partition P={{0},{−1,1},{−2,2},{−3,3},{−4,4},...} of Z. Describe the equivalence relation whose equivalence classes are the elements of P. I'm struggling to grasp what the question is asking and how to go about solving it.

Answer 1

Hint: What are the equivalence classes of the relation $\sim$ on $\mathbb Z$ defined by $x\sim y$ if and only if $x=\pm y$?

Answer 2

Consider set of integers $\Bbb Z$ With given any partition p then $x=\pm y$ if and only if {${x,y}$} $\in$ P

Answer 3

Given a set $A$ and a partition $P \subseteq \mathcal{P}(A)$ of $A$, define the relation $x \sim y$ if and only if $x$ and $y$ belongs to the same element of $P$, for all $x$ and $y$ in $A$. This is defined for all elements of $A$ since $P$ "covers" $A$, and you need to verify that this is an equivalence relation using the properties of a partition.

Answer 4

Given two sets $X$, $Y$ and a function $f:\>X\to Y$ the setting $$x\sim x'\quad:\Leftrightarrow\quad f(x)=f(x')$$ defines an equivalence relation on $X$: Two points $x$, $x'\in X$ are equivalent if $f$ assumes the same value in the two points.

In the case at hand the base set is ${\mathbb Z}$, and as $f$ you can take $f(x):=x^2$. Two integers are equivalent if their squares agree. The different values of $f$ are $0$, $1$, $4$, $9$, $\ldots$, and to each such value corresponds an equivalence class $\{0\}$, $\{-1,1\}$, $\{-2,2\}$,$\{-3,3\}$, $\ldots\ $ .