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. neither S nor T is an equivalence relation on R
B. both S and T are equivalence relations on R
C. S is an equivalence relation on R but T is not
D. T is an equivalence relation on R but S is not

Answer

A

Explanation

Since, `(1,2) in" S but "(2,1)in S`
So, S is not symmetric.
Hence, S is not an equivalence relation.
Given, `" "T={(x,y):(x-y)in l}`
Now, xTx as `x-x=0 in l`, so it is reflexive relation.
Again, let xTy `implies(x-y) in l`
`implies" "y-x in limplies"yTx"` , so T is symmetric relation.
Let xTy and yTz
`:." "x-y=l_(1)" and "y-z=l_(2)`
Now, `x-z=(x-y)+(y-z)=l_(1)+l_(2)inl`
`implies" "x-zinlimplies"xTz"`
So, T is transitive.
Hence, T is an equivalence relation.