Let R be the real line. Consider the following subsets of the plane `RxxR` . `S""=""{(x ,""y)"":""y""=""x""+""1""a n d""0""<""x""<""2},""T""=""{(x ,""y)"":""x-y""` is an integer }. Which one of the following is true?

Options

A. S is an equivalence relation on R but T is not
B. T is an equivalence relation on R but S is not
C. Neither S not T is an equivalence relation on R
D. Both S and T are equivalence relation on R

Answer

B

Explanation

For any `x in (0,2)`, we find that `x ne x + 1`.
`So, (x, x) cancelin S`.
`therefore` S is not an equivalence relation.
Reflexivity of T: For any `x in R`, we have
`x - x = 0`, which is an intege.
`implies (x, x) in T`
`therefore` T is reflexive on R
Symmetry of T: Let (x,y) `in T`. Then,
x - y is an integer
implies y - x is an integer
`implies (y,x) in T`
`therefore T` is symmetric on R.
Transitivity of T: Let (x, y) `in T` and (y, z) `in T`. Then,
x-y is an integer and y-z is an integer
implies x-z is an integer
`implies (x,z) in T`
`therefore` T is transitive on R.