NIMCET 2023 Question Paper with solutions

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Question : 49 of 120

Marks: +1, -0

Let

{\A}

and

{\B}

be two sets such that

{\A} ∩ {\X}=

{\B} ∩ {\X}=Φ

and

{\A} ∪ {\X}= {\B} ∪ {\X}

for some set

{\X}

. Then:

$A ∪ B=X$
$B=X$
$A=B$
$A=X$

Solution:

Let

A

and

B

two sets such that

\; \;\; {\A} ∩ {\X}= {\B} ∩ {\X}=φ

and

{\A} ∪ {\X}= {\B} ∩ {\X}

for some set

{\X}

To show:

{\A}= {\B}

\; \;\; {\A}= {\A} ∩( {\A} ∪ {\X})

\;= {\A} ∩( {\A} ∪ {\X})( {\A} ∪ {\X}= {\B} ∪ {\X})

\;=( {\A} ∩ {\B}) ∪( {\A} ∩ {\X})

(Distributive law)

\;=( {\A} ∩ {\B}) ∪ φ( ∵ {\A} ∩ {\X}=φ)

\;= {\A} ∩ {\B}

.....(i)
Now,

{\B}= {\B} ∩( {\B} ∪ {\X})

\;= {\B} ∩( {\A} ∪ {\X})( ∵ {\A} ∪ {\X}= {\B} ∪ {\X})

\;=( {\B} ∩ {\A}) ∪( {\B} ∩ {\X})

(Distributive law)

\;=( {\B} ∩ {\A}) ∪ φ( ∵ {\B} ∩ {\X}=φ)

\;= {\B} ∩ {\A}= {\A} ∩ {\B}

...(ii)
Hence, form (i) and (ii), we get,

A=B

.

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