Question

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 ?

• 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

Answer 1

Answer: (A)

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