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

Explanation

Given, `AnnX = BxxX = phi " … (i)"`
and `AuuX = BuuX " … (ii)"`
From Eq. (ii), `Ann(AuuX)=Ann(BuuX)`
`impliesA=(AnnB)uu(AnnX)" "[becauseAsubeAuuXthereforeAnn(AuuX)=A]`
`impliesA=(AnnB)uuphi" "[becauseAnnX=phi]`
`implies A=(AnnB)`
`implies AsubeB" ... (iii)"`
Again, `AuuX=BuuX`
`implies Bnn(AuuX)=Bnn(BuuX)`
`implies (BnnA)uu(BnnX)=B" "[becauseBsubeBuuXthereforeBnn(BuuX)=B]`
`implies(BnnA)uuphi=B" "[becauseBnnX=phi]`
`implies BnnA=B`
`implies BsubeA" ... (iv)"`
From Eqs. (iii) and (iv), we have A = B.