Show that $D$ is a distance function.

Question

I need help with the following question.

Let ($M, d$) be a metric space and $x$ ∈ $M$.

Define $D : M × M → \mathbb R$

by $D(p, q)$ = $d(p, x) + d(q, x)$ if $p$ does not equal $q$

and

$D(p, q)$ = $0$ if $p=q$

Show that $D$ is a distance function.

My attempt:

I am aware that this function holds for non-negativity when $p$ does not equal $q$

I am also aware that it holds for identification condition as the function = $0$ when $p = q$

How would i show symmetry and and that it holds for the triangle inequality?

Answer 1

Since $d$ is a metric so $d(p,x)=d(x,p)$ and $d(x,q)=d(q,x)$.

$D(p,q)= d(p,x)+d(q,x)=d(x,p)+d(x,q)= D(q,p)$

$D(p,r)= d(p,x)+d(r,x)\le d(p,r)+d(r,x)+ d(r,q)+ d(q,x)=D(p,q)+D(q,r)$