Home
Class 12
MATHS
Let A and B be sets. If A ∩ X = B ∩ X = ...

Let A and B be sets. If `A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X` for some set X, show that `A = B`.

Text Solution

Verified by Experts

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.
Promotional Banner

Similar Questions

Explore conceptually related problems

Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C . Show that B = C .

For any sets A and B, show that P ( A ∩ B ) = P ( A ) ∩ P ( B ) .

Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ .

Show that for any sets A and B, A = ( A ∩ B ) ∪ ( A – B ) and A ∪ ( B – A ) = ( A ∪ B )

Using properties of sets, show that A ∪ ( A ∩ B ) = A

Using properties of sets, show that A ∩ ( A ∪ B ) = A .

Let A and B be two non- empty subsets of a set X such that A is not a subset of B .Then

Let A and B be two sets such that n (A) =3 and n(B) =2, If ( x,1) ,( y ,2) (z,1) are in A xxB find A and B where x,y,z, are distinct elements

The negation of the statement "If x in A nu B then x in A and x in B " is

The negation of the statement "If x in A uu B then x in A and x in B " is