1.

A intersection (B union C)

Answer» \bold{Answer :}

We need to PROVE that,

A ∩ (B ∪ C) = (A ∩ B) U (A ∩ C)

LET, X ∈ {A ∩ (B ∪ C)}

⇒ (x ∈ A) ∧ {x ∈ (B ∪ C)}

⇒ (x ∈ A) ∧ { (x ∈ B) ∨ (x ∈ C) }

⇒ {(x ∈ A) ∧ (x ∈ B)} ∨ {(x ∈ A) ∧ (x ∈ C)}

⇒ {x ∈ (A ∩ B)} ∨ {x ∈ (A ∩ C)}

⇒ x ∈ {(A ∩ B) U (A ∩ C)}

∴ A ∩ (B ∪ C) ⊆ (A ∩ B) U (A ∩ C) ...(i)

Again, let y ∈ {(A ∩ B) U (A ∩ C)}

⇒ {y ∈ (A ∩ B)} ∨ {y ∈ (A ∩ C)}

⇒ {(y ∈ A) ∧ (y ∈ B)} ∨ {(y ∈ A) ∧ (y ∈ C)}

⇒ (y ∈ A) ∧ { (y ∈ B) ∨ (y ∈ C) }

⇒ (y ∈ A) ∧ {y ∈ (B ∪ C)}

⇒ y ∈ {A ∩ (B ∪ C)}

∴ (A ∩ B) U (A ∩ C) ⊆ A ∩ (B ∪ C) ...(ii)

∴ From (i) and (ii), we get

A ∩ (B ∪ C) = (A ∩ B) U (A ∩ C)

Hence, PROVED.

#\bold{MarkAsBrainliest}


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