Are there Bernouilli random variables $X_1,...,X_k$ such that these variables are negatively correlated but $1-X_1,...,1-X_k$ are not?

Question

I was reading about correlation inequalities in Alon-Spencer's book but so far I haven't found a construction such that this is the case. My guess is that since $$\rho_{ij} = {\mbox{Cov}(X_i,X_j)\over \mbox{Var}[X_i] \ \mbox{Var}[X_j]}$$ hence we want to find $X_i$ such that $$\mathbb{E}[XY] < \mathbb{E}[X]\mathbb{E}[Y]$$ but $$\mathbb{E}[(1-X)(1-Y)] < \mathbb{E}[1-X]\mathbb{E}[1-Y]$$ is not true.

Answer 1

$$Cov(1-X,1-Y) = Cov(1,1) - Cov(X,1) - Cov(1,Y) + Cov(X,Y) = Cov(X,Y)$$