= Solution
A metric dissimilarity $d$ must satisfy nonnegativity, separation, symmetry and the triangle inequality:
$$
\boxed{d(x,y)\geq0;\quad d(x,y)=0\iff x=y;\quad d(x,y)=d(y,x);\quad d(x,z)\leq d(x,y)+d(y,z).}
$$
The underlying objects must be specified: here they are binary measurement profiles, rather than labels attached to individuals.
Let $n_{11},n_{00},n_{10},n_{01}$ count coordinate pairs of the indicated types. The <simple matching coefficient> is $S=(n_{11}+n_{00})/4$, so its complementary dissimilarity is $(n_{10}+n_{01})/4$. Counting mismatches in every pair gives the <dissimilarity matrix>
$$
\boxed{D_S=\begin{pmatrix}
0&3/4&1/4&3/4\\
3/4&0&1/2&1/2\\
1/4&1/2&0&1/2\\
3/4&1/2&1/2&0
\end{pmatrix}.}
$$
For the <Jaccard coefficient>, joint absences are excluded: $J=n_{11}/(n_{11}+n_{10}+n_{01})$. Thus the <Jaccard distance> is $1-J$, giving
$$
\boxed{D_J=\begin{pmatrix}
0&3/4&1/4&3/4\\
3/4&0&1/2&2/3\\
1/4&1/2&0&1/2\\
3/4&2/3&1/2&0
\end{pmatrix}.}
$$
Only the pair of individuals two and four has a joint absence. Their profiles have one shared presence, two mismatched presences and one joint absence, giving dissimilarities $2/4$ and $2/3$, respectively. All other pairs have a presence in at least one profile at every coordinate, so the two denominators coincide.
For the requested <metric property of simple matching dissimilarity>, write a general pair of $p$-coordinate binary profiles as $x,y$. Then
$$
d_S(x,y)=1-S(x,y)=\frac1p\sum_{j=1}^p\mathbf1_{\{x_j\ne y_j\}},
$$
which is the <normalized Hamming distance>. It is nonnegative and symmetric, and equals zero exactly when all coordinates match. For each coordinate, a mismatch between $x_j$ and $z_j$ forces a mismatch between $x_j,y_j$ or between $y_j,z_j$. Therefore
$$
\mathbf1_{\{x_j\ne z_j\}}\leq\mathbf1_{\{x_j\ne y_j\}}+\mathbf1_{\{y_j\ne z_j\}}.
$$
Summing and dividing by $p$ proves the triangle inequality, establishing all four metric properties. \b[Simple matching dissimilarity is a metric on binary profiles.] If two separately labelled individuals have identical profiles, it gives zero distance between them, so on those labels it is only a <pseudometric>. The four profiles here are distinct, so this distinction does not prevent $D_S$ from defining a metric on the given sample.
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