Ireland_2017

We say that a set $X$ of boys is separated from a set $Y$ of girls if no boy in $X$ knows any girl in $Y$. Similarly, a set $Y$ of girls is separated from a set $X$ of boys if no girl in $Y$ knows any boy in $X$. Since acquaintance is mutual, separation is symmetric: $X$ is separated from $Y$ if and only if $Y$ is separated from $X$.

Let $n$ denote the number of ordered pairs $(X,Y)$ such that $X$ is a subset of boys, $Y$ is a subset of girls, and $X$ is separated from $Y$.

For any set $X$ of boys, denote by $Y(X)$ the set of girls who are not acquainted with any boy in $X$. Then $X$ is separated from exactly $2^{|Y(X)|}$ sets of girls, and so $$n=\sum _{X\subset B}{2}^{|Y(X)|}.$$ Exactly those terms in the sum are odd for which $Y(X)$ is empty, i.e., when $X$ is sociable. Therefore $n$ is congruent modulo $2$ to the number of sociable sets of boys.

A similar argument shows that $n$ is congruent modulo $2$ to the number of sociable sets of girls. Therefore, if the number of sociable sets of boys is odd, then the number of sociable sets of girls is also odd.