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 one of the following is true?

• A. Neither S nor T is an equivalence relation on R
• B. Both S and T are equivalence relation on R
• C. S is an equivalence relation on R but T is not
• D. T is an equivalence relation on R but S is not

Answer 1

Answer: (D)

Given S = {(x , y) : y = x + 1 and 0 < x < 2}
$\because$ x $\neq$ x + 1 for any x $\in$(0, 2)
$\Rightarrow $ (x, x) $\notin$ S
$\therefore $ S is not reflexive.
Hence S in not an equivalence relation.
Also T ={x, y): x - y is an integer}
$\because$ x - x = 0 is an integer $ \forall \, x \in R$
$\therefore $ T is reflexive.
If x - y is an integer then y - x is also an integer
$\therefore $ T is symmetric
If x - y is an integer and y - z is an integer then
(x - y) + (y- z) = x - z is also an integer.
$\therefore $ T is transitive