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 false, statement 2 is true.
B. Statement 1 is ture, statement 2 is true, statement 2 is a correct explanation for statement 1.
C. Statement 1 is true, statement 2 is true, statement 2 is not a correct explanation for statement 1.
D. Statement 1 is false, statement 2 is false.

Answer

D

Explanation

Statement 1 is true. We observe that
R is reflexive `xRx " as " x-x=0` is an integer `AA x in A`
Let `(x,y) in A`
`implies y-x` is an integer
`implies x-y` is also an integer
`implies R ` is Symmetric
Let `(x,y) in A and (y,z) in A`
`implies y-x` is an integer and `z-y` is an integer
`implies y-x+z-y` is also integer
`implies z-x` is an integer
`implies (x,z) in A`
`implies R` is Transitive
Because of the above properties A is an equivalence relation over R
Statement 2 is false as 'B' is not symmetric on R we observe that
`0 B x " as "0 =0 AA x in R " but " (x,0) notin B`