For πΌββ let βπΌβ denote the greatest integer smaller than or equal to πΌ. Define π:βΓββ[0,β) by π(π₯,π¦)=β|π₯βπ¦|β, π₯,π¦ββ. Then which of the following are true?

d(x,y)=0, if and only if x=y, x,yββ

What can you say about d(0.2,0.4)?

d(x,y)=d(y,x), x,yββ

Clearly, since |x – y| = |y – x|, we can be sure that this option is true.

d(x,y) β€ d(x,z) + d(z,y), x,y,zββ

What happens if x = 1.2, y = 2.2, z = 2?

d is not a metric on β

Verifying option 3, triangular inequality is not satisfied

*CSIR-NET December 2019, Part C Question.*

**Do you have any other ways to prove/disprove options above? **

Beauty of the entire question is that you simply need to understand the definition of metric spaces well to find the answer. Watch the video below to learn more.

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