Consider the following problem:
Let $X$ be a set of infinite cardinality,then the set $M(X)$ of all possible metric functions on $X$ has cardinality of $2^X$.
Proof: Suppose $\scr S\subset$$ \mathbb 2^X$ ,where $\scr S$ is a set containing all non-empty non-singleton subsets of $X$ and satisfies the property that if $A\in \scr S$,then $X-A\notin \scr S$.For each $A\in \scr S$ define $d_A$ as follows:
$d_A(x,y) = \begin{cases} 0, & \text{if $x=y$} \\[3ex] 1, & \text{if both $x,y$ are in $A$ or both in $X-A$ and $x\neq y$}\\[3ex] 2,& \text{if only one of $x$ and $y$ is in $A$}\end{cases}$
Verify that it is a metric for each $A\in \scr S$.
Also see that $A\mapsto d_A$ is an injective map from $\scr S $$\to M(X)$.So,$|M(X)|\geq |\scr S|$$=|2^X|=2^{|X|}$.
Also $|M(X)|\leq |\mathbb R|^{|X\times X|}=c^{|X|}=2^{\aleph_0 .|X|}=2^{|X|}$.
So,$M(X)$ has the cardinality of $2^X$.
Is the solution correct?Can someone give an alternative solution?