query: How to prove $R$ is an equivalence relation where: $\forall x, y \in A$, $xRy \iff 4|(x − y)$
positive_passages.docid: 18018
positive_passages.text: Proving that $4 \mid m - n$ is an equivalence relation on $\mathbb{Z}$
negative_passages.docid: 46106
negative_passages.text: Prove that $D_r \circ D_s = \{ (x,y) \in \mathbb{R}^2 \ | \ |x-y|<r+s \}$, where $D_a = \{ (x,y) \in \mathbb{R}^2\ | \ |x-y| < a \}$
negative_passages.docid: 57418
negative_passages.text: How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$
negative_passages.docid: 114198
negative_passages.text: Showing that $y$ is irreducible in $\mathbb R[x,y]/(x^2+y^2-1)$.
negative_passages.docid: 138436
negative_passages.text: How to prove reflexivity, symmetry and transitivity for the following relation?
negative_passages.docid: 50833
negative_passages.text: Find all $f:\mathbb R\to\mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $xf(y)+yf(x)=(x+y)f(x)f(y)$.
negative_passages.docid: 158009
negative_passages.text: Why is it always the case that, given an equivalence relation R on a set X and a $x \in X$, that x is equivalent to itself?
negative_passages.docid: 89167
negative_passages.text: Cardinality of equivalence relations in $\mathbb{N}$
negative_passages.docid: 37924
negative_passages.text: Equivalence of norms proof
negative_passages.docid: 131155
negative_passages.text: $x\sim y\iff x^2\equiv y^2\pmod{\!6}$ is an equivalence relation.
negative_passages.docid: 89125
negative_passages.text: Prove that $r \in R$ such that $0 \lt r \lt 1$ then $\frac{1}{r (1 - r)} \ge 4$
negative_passages.docid: 99911
negative_passages.text: How to show that if $x=(p,q)$ and $y=(p',q')$ then $x \sim y $ if $pq'=p'q$ is an equivalence relation?
negative_passages.docid: 137921
negative_passages.text: Reflexivity, Transitivity, Symmertry of the square of an relation
negative_passages.docid: 37569
negative_passages.text: Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
negative_passages.docid: 4610
negative_passages.text: Proving an identity with a combinatorial proof: $\binom{n}{k}\binom{k}{r}=\binom{n}{r}\binom{n-r}{k-r}$
negative_passages.docid: 16548
negative_passages.text: Prove $(x+y) \text{ mod } n = ((x \text{ mod } n)+(y \text{ mod } n)) \text{ mod } n,\,$ and product analog
negative_passages.docid: 139979
negative_passages.text: Proof there is a rational between any two reals