For all sets A, B and C, show that (A – B) ∩ (A – C) = A

For all sets A, B and C, show that (A – B) ∩ (A – C) = A – (B ∪ C)

CBSE | Class 11 | Maths | NCERT Exemplar | Sets

For all sets A, B and C, show that (A – B) ∩ (A – C) = A – (B ∪ C)

Solution:

As per the question,

A, B and C are three sets

We need to show:

$(\mathrm{A}-\mathrm{B}) \cap(\mathrm{A}-\mathrm{C})=\mathrm{A}-(\mathrm{B} \cup \mathrm{C})$

$\text { Let's say } \mathrm{x} \in(\mathrm{A}-\mathrm{B}) \cap(\mathrm{A}-\mathrm{C})$

$\Rightarrow \mathrm{x} \in(\mathrm{A}-\mathrm{B}) \text { and } \mathrm{x} \in(\mathrm{A}-\mathrm{C})$

$\Rightarrow(\mathrm{x} \in \mathrm{A} \text { and } \mathrm{x} \notin \mathrm{B}) \text { and }(\mathrm{x} \in \mathrm{A} \text { and } \mathrm{x} \notin \mathrm{C})$

$\Rightarrow \mathrm{x} \in \mathrm{A} \text { and }(\mathrm{x} \notin \mathrm{B} \text { and } \mathrm{x} \notin \mathrm{C})$

$\Rightarrow \mathrm{x} \in \mathrm{A} \text { and } \mathrm{x} \notin(\mathrm{B} \cup \mathrm{C})$

$\Rightarrow \mathrm{x} \in \mathrm{A}-(\mathrm{B} \cup \mathrm{C})$

$\Rightarrow(\mathrm{A}-\mathrm{B}) \cap(\mathrm{A}-\mathrm{C}) \subset \mathrm{A}-(\mathrm{B} \cup \mathrm{C}) \ldots(\mathrm{i})$

$\text { Let's say } \mathrm{y} \in \mathrm{A}-(\mathrm{B} \cup \mathrm{C})$

$\Rightarrow \mathrm{y} \in \mathrm{A} \text { and } \mathrm{y} \notin(\mathrm{B} \cup \mathrm{C})$

$\Rightarrow \mathrm{y} \in \mathrm{A} \text { and }(\mathrm{y} \notin \mathrm{B} \text { and } \mathrm{y} \notin \mathrm{C})$

$\Rightarrow(\mathrm{y} \in \mathrm{A} \text { and } \mathrm{y} \notin \mathrm{B}) \text { and }(\mathrm{y} \in \mathrm{A} \text { and } \mathrm{y} \notin \mathrm{C})$