Prove Absorption Law: A U (A and B) = A

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SUMMARY

The discussion focuses on proving the Absorption Law in set theory, specifically the equation A ∪ (A ∩ B) = A. The user attempts to demonstrate that the right-hand side is a subset of the left-hand side but struggles with the proof. They successfully show that the left-hand side is a subset of the right-hand side by defining a new set G that contains elements in both A and B, concluding that G ∪ A equals A. The user seeks clarification on how to prove the right-hand side of the equation.

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  • Understanding of set theory concepts, particularly union and intersection.
  • Familiarity with subset definitions and properties.
  • Basic knowledge of Boolean Algebra principles.
  • Experience with formal mathematical proof techniques.
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Homework Statement



Prove the absorption law A U (A and B) = A

Homework Equations



The Attempt at a Solution



I need to prove the sets are equal.

I don't know how to prove the right hand as a subset of the left hand.
For left hand as subset of right hand, it's pretty simple (we can define a new called G holding x in A and in B), and G is subset of A, then G U A is A (definition of subset and Union).

But how do I prove the right hand?
I started writing "There exists a set A..." and I thought about using partition, but then I couldn't go on anymore. If I say there exists an arbitrary set B, and x in A and B, then I am again writing the first proof.

Can someone please help me? Thanks.
 
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Is this set theory or Boolean Algebra? Boolean Algebra I think.
 

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