Question

Let A, B, and C be the sets such that A U B =A U C and A n B = A n C. Show that B=C.
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Answer 1

Answer:

let A = ( 1,2,3 ) , B = C = ( 4,5,6 )

Step-by-step explanation:

A U B = ( 1,2,3 ) U (4,5,6)

A U B = (1,2,3,4,5,6) ( first equation )

A U C = ( 1,2,3 ) U ( 4,5,6 )

A U C = ( 1,2,3,4,5,6 ) ( second equatiion )

first equation = second equation

A n B = ( 1,2,3 ) n ( 4,5,6)

A n B = ( ) ( third equation)

A n C = ( 1,2,3 ) n (4,5,6 )

A n C = ( ) ( fourth equation )

third equation = fourth equation

Only if B = C then the given operations will be clear. hence B = C

Answer 2

Answer:

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Step-by-step explanation:

Step-1:Prove using suitable formula of sets.

It is given that,

A∪B = A∪C …(1)

A∩B = A∩C…(2)

Taking  ’∩ C’ on both sides in equation (1)

(A∪B)∩C = (A∪C)∩C

We know that,

(A∪B)∩C = (A∩C)∪(B∩C) and (A∪C)∩C = C

So,

(A∩C)∪(B∩C)=C

(A∩B)∪(B∩C)=C…(3)[From(2))

Again,

Taking  ’∩ B’ on both side in equation (1)

(A∪B)∩B = (A∪C)∩B

B = (A∩B)∪(C∩B)

B = (A∩B)∪(B∩C)

B = C[From (3)]

Hence, proved.