Question

prove that : (a) (1) a c ( a u b) (2) b c (a u b) (b) (1) (a intersection b) c a (2) (a intersection b ) c b​

Answer 1

$\large\underline{\sf{Solution-}}$

1. To Prove :-

$\bf :\longmapsto\:A \: \sub \: A \: \cup \: B$

Proof :-

$\rm :\longmapsto\:Let \: x \: \in \: A$

$\rm :\implies\:x \: \in \: A \: \cup \:B$

$\bf\implies \:A \: \sub \:A \: \cup \:B$

Hence, Proved

2. Prove that

$\bf :\longmapsto\:B \: \sub \: A \: \cup \: B$

Proof :-

$\rm :\longmapsto\:Let \: x \: \in \: B$

$\rm :\implies\:x \: \in \: A \: \cup \:B$

$\bf\implies \:B \: \sub \:A \: \cup \:B$

Hence, Proved

3. Prove that

$\bf :\longmapsto\:A \: \cap \:B \: \sub \:A$

Proof :-

$\rm :\longmapsto\:Let \: x \: \in \: A \: \cap \:B$

$\rm :\implies\:x \: \in \: A \: \: and \: \: x \: \in \: B$

$\rm :\implies\:x \: \in \: A$

$\bf\implies \:A \: \cap \:B \: \sub \:A$

Hence, Proved

4. Prove that

$\bf :\longmapsto\:A \: \cap \:B \: \sub \:B$

Proof :-

$\rm :\longmapsto\:Let \: x \: \in \: A \: \cap \:B$

$\rm :\implies\:x \: \in \: A \: \: and \: \: x \: \in \: B$

$\rm :\implies\:x \: \in \: B$

$\bf\implies \:A \: \cap \:B \: \sub \:B$

Hence Proved

Additional Information :-

$\green{\boxed{ \bf{ \: A \: - \: B \: = \: A \: \cap \: {B}^{c} }}}$

$\green{\boxed{ \bf{ \: (A - B)\cup \:(A \: \cap \:B) = A}}}$

$\green{\boxed{ \bf{ \: {(A \: \cup \:B)}^{c} } = {A}^{c} \: \cap \: {B}^{c} }}$

$\green{\boxed{ \bf{ \: {(A \: \cap \:B)}^{c} } = {A}^{c} \: \cup \: {B}^{c} }}$

$\green{\boxed{ \bf{ \: A \: \cap \: {A}^{c} } = \: \phi}}$

$\green{\boxed{ \bf{ \: A \: \cup \: {A}^{c} } = \: U}}$

$\green{\boxed{ \bf{ \: { \phi}^{c} } = U}}$

Answer 2

Answer:

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