Is Min Function a Valid Metric on Cartesian Product Spaces?

[Rederick]

Homework Statement

Let (X, d1) and (Y,d2) be metric space. Define a function d:(X x Y)x(X x Y) to R by d((x1,y,1), (x2,y2))=min{d(x1,x2),d(y1,y2)}. Is d a metric on X x Y? Explain

Homework Equations

N/A

The Attempt at a Solution

Is it enough to say that the min for d((x1,y1),d(x2,y2)) is the distance function between two points and since the distance function is the minimum distance between 2 points and a metric, then d is a metric?

 

[mighty2000]

Yes, that is a valid explanation. You can also mention that d satisfies the three properties of a metric:

1. Non-negativity: d((x1,y1), (x2,y2)) = min{d(x1,x2),d(y1,y2)} ≥ 0 for all (x1,y1),(x2,y2) ∈ X x Y.

2. Symmetry: d((x1,y1), (x2,y2)) = min{d(x1,x2),d(y1,y2)} = d((x2,y2), (x1,y1)) for all (x1,y1),(x2,y2) ∈ X x Y.

3. Triangle inequality: d((x1,y1), (x3,y3)) = min{d(x1,x3),d(y1,y3)} ≤ d((x1,y1), (x2,y2)) + d((x2,y2), (x3,y3)) for all (x1,y1),(x2,y2),(x3,y3) ∈ X x Y.