boolean algebra問題

Given any Boolean function f(x1,...xi,...,xn) in switching algebra,
what is the smallest (in terms of onset sizes) Boolean function g(x1,...xi,...xn)
such that g(x1,...,,0,...,xn) = g(x1,...,,0,...,xn)and (f and~g) = 0? Express g using f, Boolean connectives (and,or,~), and/or quantifers (for all ,exist).

ps h=for all xi(f(x1,...xi,...,xn))=f(x1,...,0,...,xn) and f(x1,...,1,...,xn)
h=exist xi(f(x1,...xi,...,xn))=f(x1,...,0,...,xn) or f(x1,...,1,...,xn)
也介是說對xi這個變數做quantifer