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Show that for any sets A and B,A = (A nn...

Show that for any sets A and B,`A = (A nnB) uu(A B )`and `A uu(B A) = (A uuB)`

Text Solution

Verified by Experts

For `A=(A capB)cup(A-B)`
`(A capB)cup(A-B)=(A capB)cup(AcapB') [becauseA-B=(A cap B')]`
`= A cap (B cup B')=A cap U = A`
Therefore, `(AcapB)cup(A-B)=A`
For `A cup(B-A)=(AcupB)`
`A cup(B-A)=A cup (B cup A')` because `B-A=(B cap A')`
`= (A cup B)cap (A cup A')`
`=(A cup B) cap U` because `A cup A'=U`
`= A cup B`
Therefore, `A cup(B-A)=(A cupB)`.
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