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Let R be the real line. Consider the following subsets of the plane R \times R :

S=\left \{ (x,y):y=x+1\; and\; 0< x< 2 \right \}

T=\left \{ (x,y):x-y\; is\;an\; integer \right \}

Which one of the following is true?

  • Option 1)

    T is an equivalence relation on R but S is not

  • Option 2)

    Neither S nor T  is an equivalence relation on R

  • Option 3)

    Both S and T are equivalence relations on R

  • Option 4)

    Sis an equivalence relation on R but T is not

 

Answers (1)

best_answer

As we learnt in

REFLEXIVE RELATION -

A relation R in A is said to be reflexive,  if a R a ,∀ a ∈ A

-

 

 

SYMMETRIC RELATION -

A relation R in A is said to be symmetric, if a R b ⇒ b R a,∀ a,b ∈ A

-

 

 

TRANSITIVE RELATION -

A relation R in A is said to be transitive, if a R b and b R c ⇒ a R c ∀ a,b,c ∈ A

-

 

 S = {y = x + 1        0 < x < 2}

T = {x - y is an integer}

For T :     (a, a) it is integer reflexive

                (a, b) (b, a) it is integer symmetric.

                (a, b) (b, c) (a, c) it is integer transitive.

So it is equivalence 

For S : x = 1,  y = 2 not equivalence. 

Correct option is 1.   

 

 

 


Option 1)

T is an equivalence relation on R but S is not

This is the correct option.

Option 2)

Neither S nor T  is an equivalence relation on R

This is an incorrect option.

Option 3)

Both S and T are equivalence relations on R

This is an incorrect option.

Option 4)

Sis an equivalence relation on R but T is not

This is an incorrect option.

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