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This note is a supplement to some recent work of R.B. Bapat on Moore-Penrose inverses of set 
inclusion matrices. Among other things Bapat 
constructs these inverses (in case of existence)
for $H(s,k)$ mod $p$, $p$ an arbitrary prime, 
$0 \le s \le k \le v-s$. Here we restrict
ourselves to $p=2$. We give conditions 
for $s,k$ which are easy to state and which ensure that the 
Moore-Penrose inverse of $H(s,k)$ mod $2$ equals its 
transpose. 
E.g., $H(s,v-s)$ mod $2$ has this property. Furthermore
${\rm Ker}\,H(s,v-s)$ mod $2$ is nonzero if $0 < 2s < v \le 3s$
and then there is a decomposition
$${\rm Ker}\, H(s,v-s) \equiv \underset
{2 \mid \binom{v-s-j } { v-2s}} {\sum _
{0 \le j \le s-1}} {\rm Im}\,H(v-s,v-j)\ {\rm mod}\ 2.$$
Also, refinements of this decomposition are given.

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