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If P, Q and R are the subsets of a set A...

If P, Q and R are the subsets of a set A, then prove that `Rxx(P^(c )uuQ^(c ))^(c )=(RxxP)nn(RxxQ)`.

Text Solution

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We know that from De-Morgan's law,
`A^(c ) nn B^(c ) = (AuuB)^(c )" … (i)"`
Replacing A by `P^(c )` and B by `Q^(c )`, then Eq. (i) becomes
`(P^(c ))^(c )nn(Q^(c ))^(c ) = (P^(c )uuQ^(c ))^(c )`
`implies PnnQ=(P^(c )uuQ^(c ))^(c )" "[because(A^(c ))^(c )=A] " "...(ii)`
`therefore Rxx(P^(c )uuQ^(c ))^(c )=Rxx(PnnQ)` [from Eq. (ii)]
`=(RxxP)nn(RxxQ)` [by cartesian product]
Hence, `Rxx(P^(c )uuQ^(c ))^(c )=(RxxP)nn(RxxQ)`
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