# Why is $A$ and $A^c \cap B$ disjoint given$A ⊆ B$? What does it look like on a venn diagram?

Can someone please explain why Why is $$A$$ and $$A^c \cap B$$ disjoint given $$A ⊆ B$$? What does it look like on a venn diagram?

My incorrect interpretation (attempt) :

$$A^c$$ means that everything that is not A which includes a portion of $$B$$. $$B$$ includes $$A$$. So the intersection shouldn't be that portion of $$B$$ that just doesn't include $$A$$? Here is my sketch:

Also a more general question, how do you get "good" at finding mutually exclusive events? Because I feel like a lot of proofs require me to use the 3rd axioms of probability which involves the disjoint event.

I need this to prove that if $$A \subset B$$ then $$P(A) \leq P(B)$$.

• The green never gets to bleed into the egg yolk. Commented Mar 5, 2022 at 7:24
• Karnaugh diagrams can, in such cases, be a good alternative to Venn diagrams. Commented Mar 5, 2022 at 7:56

The fact that $$A\subseteq B$$ is not important. Since $$P\equiv A^c\cap B\subseteq A^c$$ and since $$A$$ and $$A^c$$ are disjoint, it is clear $$A$$ and $$P$$ are disjoint:
$$A\cap P\subseteq A\cap A^c=\emptyset .$$