Metric Spaces

Let d be a metric on X. Then the symmetry property of d is

• $d\left(x,y\right)\ge0$

• $d\left(x,y\right)=d\left(y,x\right)$

• $d\left(x,y\right)=0\$ if and only if $x=y$

The usual metric d on R is given by

• $d\left(x,y\right)=x+y$

• $d\left(x,y\right)=\left|x-y\right|$

• $d\left(x,y\right)=\left|x+y\right|$

Te metric defined by   $d\left(x,y\right)=0\ if\ x=y\ and\ 1\ if\ x\ne y$  is 

• indiscrete 

• discrete

• serpinski

• usual

The distance function in usual sense is

• a metric

• not a metric

In Real analysis R is equipped with which metric

• discrete

• indiscrete

• usual

• cannot say unless the topology is mentioned

Which metric is discussed in the this figure? ( may be more than one answer is correct)

• usual metric on R

• discrete metric on R

• distance metric

• Indiscrete metric on R

The triangle inequality of a metric d says

• $d\left(x,y\right)=d\left(y,z\right)$

• $d\left(x,y\right)=d\left(x,z\right)+d\left(y,z\right)$

• $d\left(x,y\right)\ge d\left(x,z\right)+d\left(z,y\right)$

• $d\left(x,y\right)\le d\left(x,z\right)+d\left(z,y\right)$

If X is a metric space with metric d on it and Y is a subset of X, then d induces a metric on X

• sometimes

• always

• never

• cannot say unless we know what d is

Euclidean plane is

• R with usual metric

• $R^2$ with usual metric

• $R^3$ with usual metric

• C with usual metriz

The attached figure demonstrate

• Usual metric in R

• Euclidean metric on plane

• $Discrete\ \ metric\ in\ R^2$

• Triangle inequality of a metric