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QJEEM08Ma2e9d: JEE Main 2008 Mathematics - Class 11 JEEM08Ma2e9d Solution: Let R be the real line. Consider the following subsets of the…

Let R be the real line. Consider the following subsets of the plane R×RR \times R :
S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right\}
T={(x,y):x−y   is  an  integer }T = \left\{ {(x,y): x - y\,\,\,is\,\,an\,\,{\mathop{\rm int}} eger\,} \right\} ,

Which one of the following is true ?

Multiple Choice Options

  • A Neither S nor T is an equivalence relation on R
  • B Both S and T are equivalence relation 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

Step-by-Step Verified Solution

Given S={(x,y):y=x+1   S = \left\{ {\left( {x,y} \right):y = x + 1\,\,} \right.\,

and \,\,\,\left. {0 < x < 2} \right\}

As     x≠x+1   \,\,\,\,x \ne x + 1\,\,\,

for any    x∈(0,2)⇒(x,x)∉S\,\,\,x \in \left( {0,2} \right) \Rightarrow \left( {x,x} \right) \notin S

∴\therefore SS is not reflexive.

Hence SS in not an equivalence relation.

Also    T={x,y):x−y\,\,\,T = \left\{ {x,\left. y \right)} \right.:x - y is an integer }\left. {} \right\}

as x−x=0x - x = 0 is an integer ∀x∈R\forall x \in R

∴\therefore TT is reflexive.

If x−yx-y is an integer then y−xy-x is also an integer

∴\therefore TT is symmetric

If x−yx-y is an integer and y−zy - z is an integer then

(x−y)+(y−z)=x−z(x-y)+(y-z)=x-z is also an integer.

∴\therefore TT is transitive

Hence TT is an equivalence relation

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