Proving AU(AΠB)=AΠ(AUB) | Intersection of Sets Proof

[neelakash]

Homework Statement

I am to prove that AU(AΠB)=AΠ(AUB)=A where Π means intersection

Homework Equations

The Attempt at a Solution

LHS xЄA or, xЄ(AΠB)
=> xЄA or, (xЄA and xЄB)
=> (xЄA or, xЄA) and (xЄA or,xЄB)
=>xЄA and xЄ(AUB).......This is precisely AΠ(AUB)

x is an element of both A and (AUB),that is what the and operation mean.
Hence xЄA......proved from one side.

Now we are to prove that yЄA =>yЄ AU(AΠB)
Since A is a subset of AU(AΠB),it follows that if yЄA,it must be true that yЄ AU(AΠB)
Again,since AU(AΠB)=AΠ(AUB),hence yЄ AΠ(AUB) too.

Hence proved.

 

[NateTG]

You certainly have the idea. This stuff is all relatively obvious, so how much you have to show depends a bit on your class.