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Let A and B be sets. If A nnX = B nnX =v...

Let A and B be sets. If `A nnX = B nnX =varphi`and `A uuX = B uuX`for some set X, show that `A = B`.(Hints `A = A nn(A uuX), B = B nn(B uuX)`and use Distributive law)

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Let sets A and B are such that, for any set X,
`A cupX=BcupX`
`rArrAcap(A cupX)=A cap(BcupX)`
`rArrAcap(A cupX)=(A cap B)cup (A cap X)` (From distributive law)
`rArrAcap(A cupX)=(AcapB)cup phi(becauseA capX=phi)`
`rArrA=(AcapB)cupphi`
`rARr A =(A capB)`...(1)
Again for any set X,
`A cup X = B cup X`
`rArr B cap (A cup X)=B cap (B cup X)`
`rArr (B cup A)cup(B cap X)=B cap (B cup X)` (From distributive law)
`rArr(B cupA)cup phi=Bcap(BcupX)[because(B capX)=phi]`
`rArr(BcupA)cupphi=B`
`rArr B=(B capA)`
`rArr B= A cap B`....(2)
From equations (1) and (2),
`rArr A =B`.
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