<s>[INST] Determine which of the following are metrics on $X$ Let $d$ and $d'$ be metrics on a non empty set $X$. Then which of the following are metrics on $X$.

*

*$\rho_1 (x,y)=d(x,y)d'(x,y)$ for all $x,y \in X$


*$\rho_2 (x,y)=max\{5,d(x,y)\}$ for all $x,y \in X$


*$\rho_3 (x,y)$=max$\{d(x,y),d'(x,y)\}$ for all $x,y \in X$


*$\rho_4 (x,y)$=min$\{d(x,y),d'(x,y)\}$ for all $x,y \in X$
My Attempt:
If $d$ satisfy the following conditions then it becomes a metric:

*

*$d(x,y) \geq 0$ for all $x$, $y \in X$.

*$d(x,y) = 0$ if and only if $x=y$.

*$d(x,y) = d(y,x)$ for all $x$, $y\in X$.

*$d(x,z) \leq d(x,y) + d(y,z)$ for all $x$, $y$, $z \in X$ (the triangle inequality)

I have seen that $\rho (x,y)=min\{5,d(x,y)\}$ for all $x,y \in X$ form a metric.
Option 3 is option that is true and options 1,2,4 are false.
But in my calculation (not written here), all the four options are correct. First three properties are holds trivially. I think there is a mistake made by me in last property I.e. triangle inequality fails for options 1,2,4. Please provide  hints or solutions. Thanks
 [/INST] Option 1 is false. For a counterexample, take $X = \mathbb R$ and let $d = d'$ be the usual metric $d(x, y) = |x - y|$. Then observe that:
$$
\rho(4, 3) + \rho(3, 2)
= |4 - 3|^2 + |3 - 2|^2
= 2
< 4
= |4 - 2|^2
= \rho(4, 2)
$$
Option 2 is false, since $\rho(7, 7) = 5$, instead of $0$.
Option 4 is false. For a counterexample, take $X = \mathbb R^2$ and let:
\begin{align*}
d(x, y) &= \sqrt{4(x_1 - y_1)^2 + (x_2 - y_2)^2} \\
d'(x, y) &= \sqrt{(x_1 - y_1)^2 + 4(x_2 - y_2)^2}
\end{align*}
Now consider $x = (1, 0)$ and $y = (0, 0)$ and $z = (0, 1)$. Observe that:
\begin{align*}
\rho(x, y) + \rho(y, z)
&= \min\left\{\sqrt{4 \cdot 1^2 + 0^2}, \sqrt{1^2 + 4 \cdot 0^2}\right\} + \min\left\{\sqrt{4 \cdot 0^2 + 1^2}, \sqrt{0^2 + 4 \cdot 1^2}\right\} \\
&= 2\\
&< \sqrt 5 \\
&= \min\left\{\sqrt{4 \cdot 1^2 + 1^2}, \sqrt{1^2 + 4 \cdot 1^2}\right\} \\
&= \rho(x, z)
\end{align*}
</s>