Negation of set statement What is the negation of the following statement?
$$B \subseteq A \Rightarrow C \subsetneq A$$
So far, I have:
$$B \subseteq A \land \neg (C \subsetneq A)$$
How do I simplify it further?
 A: The expression $\neg(C\subsetneqq A)$ says that $C$ is not a proper subset of $A$. There are therefore two possibilities: either $C$ is not a subset of $A$ at all; or $C$ is a subset of $A$ but not a proper one, in which case $C=A$. Thus, the expression is equivalent to
$$C\nsubseteq A\lor C=A\;.$$
Your whole expression is therefore equivalent to
$$B\subseteq A\land\big(C\nsubseteq A\lor C=A\big)\;.\tag{1}$$
You can apply distributivity of $\land$ over $\lor$ to get
$$(B\subseteq A\land C\nsubseteq A)\lor B\subseteq A=C\;.$$
Whether either of these is actually simpler than $B\subseteq A\land\neg(C\subsetneqq A)$ is a matter of taste, opinion, and how you’re intending to use the expression; there isn’t any really nice equivalent expression. Expression $(1)$ is arguably the clearest, since it shows explicitly the two ways in which $C$ can fail to be a proper subset of $A$.
A: [Note: Following comments, this answer assumes $\subsetneq$ means $\not\subseteq$, or not a subset. Apparently $\subsetneq$ is undefined.]
Let $P=B\subseteq A$ and $Q=C\subsetneq A$. Then the negation of $P\Rightarrow Q$ is $P\not\Rightarrow Q$, which is called Material nonimplication, and has the logical equivalence $P\land \lnot Q$.
\begin{array} {c|c|c|c}
P&Q&P\Rightarrow Q&P\not\Rightarrow Q\\
\hline
T&T&T&F\\
T&F&F&T\\
F&T&T&F\\
F&F&T&F
\end{array}
In your case, $C \subsetneq A$ has the simple negation to $C \subseteq A$, so we arrive at:
$$B \subseteq A\land C \subseteq A\to B \cup C \subseteq A$$
