Let A and B be any two sets. If for some set X, `AnnX = BnnX = phi and AuuX = BuuX`, prove that A = B.

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.