What is the negation of "$A\subseteq B$"? I'm trying to prove something using an indirect proof, so I need to know the negation of $A\subseteq B$. I'm assuming it's $A\nsubseteq B$, but what does this mean symantically? Is it $\forall x\in A,\ x\notin B$? Is it safe to say that $A$ and $B$ are disjoint at that point?
 A: $$A \subseteq B \equiv \forall x( x \in A \rightarrow x \in B)\tag{1}$$
Negating $(1)$ gives us:
$$
\begin{align} A \not\subseteq B & \equiv \lnot \forall x(x\in A \rightarrow x \in B) \\ \\
& \equiv \exists x \lnot(x\in A \rightarrow x \in B) \\ \\
& \equiv \exists x \lnot[\lnot(x \in A) \lor (x \in B)] \tag{$p \to q \equiv \lnot p \lor q$}\\ \\
& \equiv \exists x [\lnot\lnot(x \in A) \land \lnot (x \in B)] \tag{DeMorgan's}\\ \\
& \equiv \exists x [x \in A \land \lnot (x \in B)] \\ \\
& \equiv \exists x (x\in A \land x \notin B) \tag{$A\not \subseteq B$}
\end{align}$$
A: Remember the the definition of subset: if $A \subseteq B$, then every element of A is an element of $B$. The negation of a statements like "every object is " or "all objects are" is the statement "at least one object is not." Thus, $A \not \subseteq B$ means that at least one element of A is not an element of $B$.
A: Hint: Use $$A\subseteq B\equiv \forall x(x\in A\Longrightarrow x\in B)$$ to see it is not as you stated.
A: To address the last part, which wasn't your main question:  The statement you have, "$\forall x\in A$, $x\notin B$" is indeed equivalent to $A$ and $B$ being disjoint.  But quite clearly $A$ not being a subset of $B$ is much weaker than $A$ and $B$ being disjoint, for example $B$ could even be a proper subset of $A$.
A: Unfortunately not. $A \subset B$ means $\forall x \in A, x \in B$, so the negation is $\exists x \in A, x \not \in B$ i.e. there is some element of $A$ that is not in $B$.
Intuititively,  since $A \subset B$ means that $A$ is "entirely contained in $B$". $A$ not being a subset of $B$ then means that $A$ is not entirely contained in $B$. This includes the situation that they are disjoint, but also includes the situation where some elements of $A$ are in $B$, but not all of them (i.e. $A$ is partially contained in $B$).
A: We can use that

*

*$A\subset B$ is equivalent to $A\cap B^c=\emptyset$
so then if you negate it:

*

*$A\not\subset B$ is equivalent to $A\cap B^c\neq \emptyset$.

That is, there exist $a\in A$ such that $a\not\in B$, which is the definition of $A\subset B$ negated. For me, the later expression is more clear. Hope it helps someone else.
