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Proposition : $P(E \cup F) = P(E) + P(F) - P(EF)$
I have the following laws and axioms available to me:

Laws from Set Theory

Commumutative Laws: $E \cup F = F \cup E$; $EF=FE$
Associative Laws: $(E \cup F) \cup G = E\cup(F \cup G)$; $(EF)G = E(FG)$
Distributive Laws: $(E \cup F)G=EG\cup FG$; $EF \cup G = (E \cup G)(F \cup G)$

DeMorgan's Laws

$(E \cup F)^c = E^cF^c$
$(EF)^c = E^c \cup F^c$

Axioms of Probability

Axiom 1: $0 \leq P(E) \leq 1$
Axiom 2: $P(S) = 1$
Axiom 3: $P\left(\bigcup_{i=1}^{n}E_i\right) = \sum_{i=1}^{n}P(E_i) $ For $E_i$ mutually exclusive

My attempt

I tried to use Axiom 3 by constructing mutually exclusive sets: Sets $(E-F), EF, (F-E)$ are mutually exclusive and we can write $$ P(E \cup F) = P(E-F)+P(EF)+P(F-E) $$ However, this requires another proposition that $P(X-Y) = P(X) - P(XY)$. The proof requires a Venn diagram. The first proposition has also been proved using a Venn diagram in the book. I am wondering if the proposition can be proved using the laws and axioms alone.

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  • $\begingroup$ Well, $X = X \cup (Y - Y) = XY \cup (X - Y),$ right? And $XY \cap (X - Y) = X \cap (Y - Y) = \emptyset.$ $\endgroup$ Jun 28, 2022 at 6:53
  • $\begingroup$ @StephenDonovan didn't understand why $X \cup (Y-Y) = XY \cup (X-Y)$, which law are you using here? $\endgroup$
    – ananta
    Jun 30, 2022 at 2:19
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    $\begingroup$ My apologies, I didn't write it out before commenting and it seems I confused myself somehow: what I should've done was $X = X \cap S = X \cap (Y \cup Y^c) = (X \cap Y) \cup (X \cap Y^c) = XY \cup (X - Y).$ It's just using the distributive property $\endgroup$ Jun 30, 2022 at 2:28
  • $\begingroup$ @StephenDonovan thank you, and it is easy to show that $XY \cap (X-Y) = \phi$ from set theory (hopefully this translates nicely to probability). Then we use the third axiom to prove the proposition. This is what I understood. If you want you can write the answer and I will accept it. $\endgroup$
    – ananta
    Jun 30, 2022 at 3:03

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