For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2} $; prove $\equiv$ is an equivalence relation.
57466
Reflexive and relation
131155
$x\sim y\iff x^2\equiv y^2\pmod{\!6}$ is an equivalence relation.
18018
Proving that $4 \mid m - n$ is an equivalence relation on $\mathbb{Z}$
15865
Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected.
11342
Show that conguence $\bmod m\,$ is an equivalence relation
85507
Equivalence Relation problem
32155
Prove that if for every $x \in \mathbb{R}^N$ $Ax=Bx$ then $A=B$
25155
Suppose $R$ and $S$ are transitive relations on $A$. Prove that if $S \circ R \subseteq R \circ S$ then $R \circ S$ is transitive.
15312
Show that $R \cap R^{−1}$ is an equivalence relation on $X$?
53101
Is it true that the relation |A| < |B| is a sufficient condition for claiming that $f$ is a bijection?
140988
Prove $\exp(x)\exp(y)=\exp(y)\exp(x)$ if $[x,y]=0$.
23946
Prove the definitions of $e$ to be equivalent
55461
How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$
9110
Let G be a group such that $(xy)^2 = (yx)^2$ for all x, y ∈ G. Show that $xy^2 = y^2x$ for all x, y ∈ G.
133228
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
104987
Equivalence Relation with dividing x and y integers
60140
Proof of Equivalence Relation $R=\{(a,b)\in\Bbb N^2: 5\mid (2a+3b)\}$