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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:

enter image description here

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 $P(A) \leq P(B)$.

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

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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:

$$A\cap P\subseteq A\cap A^c=\emptyset .$$

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