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Home » Using properties of sets prove the statements given in Exercise for all sets A and B, A – (A – B) = A ∩ B.
Using properties of sets prove the statements given in Exercise for all sets A and B, A – (A – B) = A ∩ B.
CBSE | Class 11 | Maths | NCERT Exemplar | Sets
Using properties of sets prove the statements given in Exercise for all sets A and B, A – (A – B) = A ∩ B.

Solution:

As per the question,

Two sets are A and B

We need to prove: A-(A-B)=A \cap B

Left Hand Side =\mathrm{A}-(\mathrm{A}-\mathrm{B})

As, A-B=A \cap B^{\prime}, we have,

\begin{array}{l} =A-\left(A \cap B^{\prime}\right) \\ =A \cap\left(A \cap B^{\prime}\right) \end{array}

As, (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}, we have,

=A \cap\left[A^{\prime} \cup\left(B^{\prime}\right)^{\prime}\right]

As, \left(B^{\prime}\right)^{\prime}=B, we have,

=A \cap\left(A^{\prime} \cup B\right)

As, distributive property of set \Rightarrow(A \cap B) \cup(A \cap C)=A \cap(B \cup C), we have,

=\left(A \cap A^{\prime}\right) \cup(A \cap B)

As, A \cap A^{\prime}=\Phi, we have,

=\Phi \cup(A \cap B)

=A \cap B

= Right Hand Side

As a result hence proved.

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