The random variables $X$ and $Y$ each take values in $\{0,1\}$, and their joint distribution $p(x, y)=P\{X=x, Y=y\}$ is given by

$$p(0,0)=a, \quad p(0,1)=b, \quad p(1,0)=c, \quad p(1,1)=d .$$

Find necessary and sufficient conditions for $X$ and $Y$ to be (i) uncorrelated; (ii) independent.

Are the conditions established in (i) and (ii) equivalent?