Let R be the set of real numbers.
Statement 1:`A={(x,y) in R xx R : y-x` is an integer} is an equivalence relation on R.
Statement 2: `B= {x,y} in Rxx R : x=alpha y` for some rational number `alpha`} is an equivalence relation on R.

Options

A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for statement-1.
B. Statement-1 is True, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True.

Answer

C

Explanation

Reflexivity: For any `x in R`, we have
x - x = 0, which is an integer
`implies (x, x) in A " for all x"in R`
implies A is reflexive relation on R.
x-y is an integer
implies y-x is an integer
`implies (y,x) in A`
`therefore` A is symmetric relation on R.
Transitivity: Let `(x, y) in A` and `(y, z) in A`. Then, x-y is an integer and y-z is an integer
implies x-z is an integer
`implies (x, z) in A`
`therefore` A is a transitive relation on R.
Hence, A is an equivalence relation on R.
So, statement-1 is true.
We have,
`0=0xx2`, where 0 is a rational number
`implies (0,2)inB`
But, `2ne alpha xx 0` for any rational number `alpha`.
Thus, (0, 2) `in B` but `(2, 0) cancelin B`.
So, B is not an equivalence relation on R.
Hence, statement-2 is not true.