Let R be the real line consider the following subsets of the plane R × R S = (x, y ): y = x + 1 and 0

Question

Let R be the real line consider the following subsets of the plane R × R S = (x, y ): y = x + 1 and 0 < x < 2 T = (x, y): x - y is an integer which of the following is true ? (A) T is an equivalence relation on R but S is not (B) Neither S nor T is an equivalence relation (C) Both S and T are equivalence relation on R (D) S is an equivalence relation but T is not

Tardigrade Admin · Accepted Answer

x R x Leftrightarrow 2x2-3x2 +x2 = 0 which is true. But x R y does not give y R x since 2y2 - 3xy + x2 may not be zero if 2x2 - 3xy + y2 = 0

Q. Let $R$ be the real line consider the following subsets of the plane $R \times R$
$S = {(x, y) : y = x + 1$ and $0 < x < 2}$
$T = {(x, y) : x - y$ is an integer $}$
which of the following is true ?

$T$ is an equivalence relation on $R$ but $S$ is not

Neither $S$ nor $T$ is an equivalence relation

Both $S$ and $T$ are equivalence relation on $R$

$S$ is an equivalence relation but $T$ is not

$x R x \Leftrightarrow 2 x^{2} - 3 x^{2} + x^{2} = 0$ which is true.
But $x R y$ does not give $y R x$
since $2 y^{2} - 3 x y + x^{2}$ may not be zero
if $2 x^{2} - 3 x y + y^{2} = 0$

Q. Let R be the real line consider the following subsets of the plane R×R S={(x,y):y=x+1 and 0<x<2} T={(x,y):x−y is an integer} which of the following is true ?

T is an equivalence relation on R but S is not

Neither S nor T is an equivalence relation

Both S and Tare equivalence relation on R

S is an equivalence relation but T is not