Can someone tell if this can be simplified further? Y = ~A + ~B*~D + BD + ~C~B + ~C*D

I have a simple question about whether or not my approach is correct in simplifying a polynomial, here it is, $(n(n+1)/2)^2 = ((n^2+n)/2)^2 = 1/4(n^4+2n^3+n^2)$ I apologize if you find that hard to read, I can't write the equations they way I would like. Many thanks.

The cartesian product $M\times N$ is complete if, and only if $M$ and $N$ are complete. My approach: Let $M$ and $N$, complete metric space, then we take a cauchy sequence $z_{n}=\{x_{n},y_{n}\}\in M\times N$. And now, the projection $p_{1}:M\times N\to M$ and $p_{2}:M\times N\to N$, are u...

Suppose $(X,d)$ is a complete metric space and $(Y,d)$ is its completion . What can be said from this $?$. I think $Y=X$ is the answer . Am I correct $?$

Completeness is an important notion in mathematics. Metric spaces can be complete, a theory can be complete, a logic can be complete, and set of propositional connectives can be complete, and many other objects can be complete. We tried to remove it already. But completeness keeps coming back. ...