If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space?

[ag2ie]

If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?

[micromass]

Well, what do you think?? What have you tried?

[ag2ie]

It's easy to verify d(x,y)=0 iff x=y and d(x, y)=d(y,x),
but I don't know how to prove triangle Inequality...

[micromass]

Well, is the following trye

d(x,z)\leq \max\{d(x,y),r(x,y)\}+\max\{d(y,z),r(y,z)\}

???

[ag2ie]

yes..but if r(x,z) is greater than d(x,z), and r(y,z) is smaller than d(y,z), then this inequality is not necessary true...right?

[micromass]

Sorry, I made a typo, check the post again.

[ag2ie]

I see...if r(x,y) is greater than d( x,y), then d(x,z)≤max{d(x,y),r(x,y)}+max{d(y,z),r(y,z)} is also true...

Thanks ...and I think (X, min(d, r)) is not a metric space..right?

[micromass]

Thanks ...and I think (X, min(d, r)) is not a metric space..right?

right.

[ag2ie]

Thanks..you are really helpful