# How to understand logical implication in Set theory?

I am reading about the wiki page of material conditional (a.k.a logical implication).

In the diagram in that page (which is also pasted here), it draws an Venne Diagram of the truth function of $A \implies B$

In the text explanation (in 2nd paragraph), it says $\ P \implies Q$ can be symbolized using set theory: $P \supset Q$

Question: The Diagram explanation and text explanation is conflicting. It seems in the diagram, A is not a superset of B. What is the correct way to understand it?

• Honestly, you don't appear to have read the caption of the image that you took from the linked wikipedia article. It explains that the white area is the set of items for which the material implication is false. Also it doesn't say that it can be symbolized using set theory, it says that the symbol can be used for implication, but also has a meaning in set theory. – jgon Dec 28 '17 at 5:51
• With the "horseshoe" symbol: $\supset$ he does not mean "superset". It is the symbol used sometimes (mainly: beginning XX Century) in place of $\to$ of $\Rightarrow$. See The Notation in Principia Mathematica. – Mauro ALLEGRANZA Dec 28 '17 at 7:36

Use the fact that $$A \to B \equiv \neg(A \wedge \neg B)$$
Note that $A \wedge \neg B$ represents the portion of the set A excluding the area of intersection of $A$ and $B$ (the white portion). The negation gives us the required red area.