TITLE: "If A then B" in Venn (or Euler) Diagrams
QUESTION [9 upvotes]: 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.

REPLY [5 votes]: 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).