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

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

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

Text Solution

Verified by Experts

The correct Answer is:
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.
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