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Let A, B and C be the sets such that A u...

Let `A`, `B` and `C` be the sets such that `A uu B=A uu C` and `A nn B = A nn C`. show that `B=C`

Text Solution

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Given, `AuuB = AuuC" … (i)"`
and `AnnB = AnnC " … (ii)"`
To prove B = C.
From Eq. (i), `(AuuB)nnC=(AuuC)nnC`
`implies(AnnC)uu(BnnC)=(AnnC)uu(CuuC)`
`implies(AnnB)uu(BnnC)=(AnnC)uuC" "[becauseAnnC=AnnB]`
`implies (AnnB)uu(BnnC)=C" " [becauseAnnCsubeC]`
`"Thus, "C=(AnnB)uu(BnnC)" "...(iii)`
Again, from Eq. (i), `(AuuB)nnB=(AuuC)nnB`
`implies (AnnB)uu(BnnB)=(AnnB)uu(CnnB)`
`implies (AnnB)uuB=(AnnB)uu(BnnC)`
`implies B=(AnnB)uu(BnnC)" "[becauseAnnBsubeB]" ... (iv)"`
Thus, `B=(AnnB)uu(BnnC)`
From Eqs. (iii) and (iv), we have B = C.
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