Prove A.(B+C) = (A.B)+(A.C) <Boolean Algebra>

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The discussion focuses on proving the Boolean algebra law A.(B+C) = (A.B)+(A.C), which states that AND is distributive over OR. Participants express frustration with the lack of algebraic proofs available online, as most resources emphasize the distributive property of OR over AND. The inquiry seeks alternative methods to validate this law without relying on truth tables or Venn diagrams. The conversation also touches on the foundational axioms necessary for such a proof. Overall, the thread highlights a gap in accessible algebraic proofs for this specific Boolean algebra law.
Valour549
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Most of the results on google happily prove A+(B.C) = (A+B).(A+C), which is that OR is distributive (over AND).

But as part of their proof, they use the law that AND is distributive (over OR), namely that
A.(B+C) = (A.B)+(A.C) which I can't seem to find any algebraic proof for.

So are there any ways to prove this law without using a truth table or venn diagram?
 
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What are the axioms you start with?
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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