Let R be the real line. Consider the following subsets of the plane R × R.
$S=\{(x,y):y=x+1\,and\,0<x<2\}$
$T=\{(x,y):x-y\,is\,an\,integer\}$
Which one of the following is true?
Answer & explanation
Correct answer: option 4
$T=\{(x,y):x-y∈Z$
As $0 ∈ zT$ is a reflexive relation
If $x - y ∈ z ⇒ y - x ∈ z$
∴ T is symmetrical also
If $x - y = z_1$ and $y-x=z_2$
Then $x-z=(x-y)+(y-z)=z_1+z_2∈ z$
∴ T is also transitive.
Hence T is an equivalence relation
Clearly $x≠x+1⇒(x,x)∈S$
∴ S is not reflexive