LetR 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 }.Whichone 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
A. Both S and T are equivalence relations on R
B. S is an equivalence relation on R but T is not
C. T is an equivalence relation on R but S is not
D. Neither S nor T is an equivalence relations on R
A. Both S and T are equivalence relations on R
B. S is an equivalence relation on R but T is not
C. T is an equivalence relation on R but S is not
D. Neither S nor T is an equivalence relations on R
Correct Answer – C
`T={(x,y):x-yinI}`
As `0 in I`, so T is a reflexive relation
If `x-yinIimpliesy-x inI`
`therefore` T is symmetric also.
If x – y = I and y – z = 12
Then, `x-z=(x-y)+(y-z)=I_(1)+I_(2)inI`
`therefore` T is also transitive.
Hence, T is an equivalence relation. Clearly, `xnex+1implies(x,x)cancelinS`
`thereforeS` is not reflexive.