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

Answer

A

Explanation

Statement-1 `A={(x,y)inRxxR:y-x" is an integer"}`
(a) Reflexive xRy : (x-x) is an integer
which is true.
Hence it is reflexive.
Symmetric xRy : (x - y) is an integer.
implies - (y - x) is also an integer.
`therefore` (y - x) is also an integer.
implies y R x
Hence, it is symmetric.
Transitive x R y and y R z
implies (x - y) and (y - z) are integere and
implies (x - y) + (y - z) is an integer.
implies (x - z) is an integer
implies x R z
`therefore` It is transitive
Hence, it is equvalence relation.
Statement-2
`B={(x,y)inRxxR:x = alphay" for some reational number "alpha}`
If `alpha = 1`, then x R y : x = y (To check equivalence)
(a) Reflexive xRx : x = x (True)
`therefore` Reflexive
(b) Symmetric `xRy:x=yimpliesy=ximpliesyRx`
`therefore` Symmetric
(c ) Transitive xRy and yRz implies x = y
and y = z implies x = z implies xRz
`therefore` Transitive
Hence, it is equivalence relation.
`therefore` Both are true but Statement-2 is not correct explanation of Statement-2