For š¼āā let āš¼ā denote the greatest integer smaller than or equal to š¼. Define š:āĆāā[0,ā) by š(š„,š¦)=ā|š„āš¦|ā, š„,š¦āā. Then which of the following are true?

d(x,y)=0, if and only if x=y, x,yāā

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

d(x,y)=d(y,x), x,yāā

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

d(x,y) ā¤ d(x,z) + d(z,y), x,y,zāā

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

d is not a metric on ā

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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