Question

Consider R is real number and S and R are subsets of R x R defines as :
Which one of the following is true ?

$S=\left\{\right(x,y):y=x+1and0<x<2$
$T=\left\{\right(x,y):x-yisaninteger\}$

• A. T is an equivalent realtion on R but S is not
• B. Neither S nor T is an equivalence relation on R
  
• C. S is an equivalence relation on R but T is not
• D. Both S and T are an equivalence relation on R

Answer 1

The correct option is A T is an equivalent realtion on R but S is not

1. S = {(x, y) :y = x + l and 0 < x < 2}
• Check for Reflexive Relation:
$(x,x):x=x+1butx\ne x+1$
Hence cannot be reflexive S is not equivalence relation on R.

2. T ={(x, y): x-y is an integer}
• Check for Reflexive Relation:
$(x,x):x-xisintegerx-x=0and0ϵinteger$
So, T is reflexive.

• Check for Symmetric Relation:
(x , y) : x - y is integer and (y, x) : y - x also an integer.
So. T is symmetric relation.

• Check to Transitive Relation:
(x - y ) : x - y is integer and (y, z) : y - z is integer then (x : z ) : x - z is also integer.
So, T is transitive.
Hence T is equivalence relation but S is not.