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Let R be the real line. Consider the f...

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?
(1) neither S nor T is an equivalence relation on R
(2) both S and T are equivalence relations on R
(3) S is an equivalence relation on R but T is not
(4) T is an equivalence relation on R but S is not

Text Solution

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`T-> x -> R`
`x-x = 0 in I`(x,x) `in`T reflexive
`(x,y) in T ; (y,x) in T`
`x-y = I; y-x in I` symmetric
if `(x,y) in T, (y in z) in T then (x,z) in z`
`y-x = I_1`
`z-y = I_2`
`z-x = I_1 + I_2 = I'`
...
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