For α in R let ⌊α⌋ denotes greatest integer less than or equal to α

Q. For  α in R  let ⌊α⌋ denotes greatest integer less than or equal to α.

Solution :

Given function

 d:R×R→[0, ∞]

defined as

 d(x,y)=⌊|x-y|⌋

For the option (1)

Suppose d(x,y)=0=> ⌊|x-y|⌋=0=>0≤|x-y|<1 ⇏x=y

For example x =0, y =0.5 then d(0,0.5)=⌊|0-0.5|⌋=⌊0.5⌋=0

Therefore option (1) is not correct.

For the option (2)

d(x,y)=⌊|x-y|⌋=⌊|y-x|⌋=d(y,x) 

So option (2) is correct.

For the Option (3)

Let’s choose x=0, y=1, z =0.5 then

d(0,1)=⌊|0-1|⌋=1

d(1,0.5)=⌊|1-0.5|⌋=⌊|0.5|⌋=0

d(0.5,0)=⌊|0.5-0|⌋=⌊|0.5|⌋=0

d(0,0.5)+d(1,0.5)=0+0=0

That is d(0,1)>d(0,0.5)+d(1,0.5)

So this option is also not true.

For the fourth option

From options (1) and (3) properties of a metric space are not satisfied Hence d(x,y) is not a metric on R. SO option (4) is true.

Therefore options (2) and (4) are correct.