"If A then B" in Venn (or Euler) Diagrams How can I represent "If A then B" in a diagram?
I thought it would be a simple subset like $A ⊂ B$. However this material says that
If $A$ then $B$ $=$ $A^c ∪ B$.
Now I am confused.
 A: You want to construct the set $\;\{x\mid x\in A \to x\in B\}\;$.
Then by implication equivalence this is $\;\{x\mid x\not\in A \vee x\in B\}\;$.
Which is simply $\;A^\complement \cup B\;$.
This is the set of all elements that, if they're in A then they're in B

The statement $A\subseteq B$ is not a set.  It is a relation.  It is the statement that $y\in A \implies y\in B$.
In the specific case that  $A$ is a subset of $B$, then there is no element that is not in $A^\complement \cup B$.

So if you wanted to represent the statement "if $A$ then $B$", you could have $A$ as a subset of $B$.
But if you wanted to represent all elements that "if in $A$ then in $B$" you would use the union: $A^\complement\cup B$.
A: Both representations you mentioned represent different things. "If $A$ then $B$" is a statement that is either true or false if each of $A,B$ are either true or false. The statement is true under certain conditions, namely as long as we do not have both $A$ being true and $B$ being false, which is equivalent to having either $A$ being false or $B$ being true, which is $(\neg A) \lor B$. Now if you represent $A,B$ by the sets of conditions where $A,B$ respectively are true, then $(\neg A) \lor B$ would be represented by the set $A^c \cup B$. This is why the venn diagram for implication is often given to have $A^c \cup B$ shaded.
On the other hand, your interpretation is not wrong either, but just about a different thing. The assertion "If $A$ then $B$" says that every condition under which $A$ is true is also a condition under which $B$ is true. Hence the set of conditions where $A$ is true is a subset of the set of conditions where $B$ is true.
So your diagram corresponds to the assertion itself (what is the case when it is true), while the other diagram corresponds to the truth value of the assertion (the shaded region is when it is true).
A: "If $A$, then $B$" can be represented logically by the complement of its negation.
Specifically, "NOT (If $A$, then $B$)" $\leftrightarrow$ "$A$ and not $B$", so that "If $A$, then $B$" is the same as "NOT ($A$ and not $B$)" which, using De Morgan's laws, is "(not A) OR B". That is, you want $A^c\cup B$, as they said.
A: It is quite simple - the external shape is B and A is fully included

