Question

वत स्या
का पूरा
(i) यदि A = {a, b}, B = {a, b, C} तब AU B =
/::)
तयार
Ra शतगत
है
XR में -​

Answer 1

Thanks for A2A...

It is called "Distributive Property" for sets.Here is the proof for that,

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Let x ∈ A ∪ (B ∩ C). If x ∈ A ∪ (B ∩ C) then x is either in A or in (B and C).

x ∈ A or x ∈ (B and C)

x ∈ A or {x ∈ B and x ∈ C}

{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}

x ∈ (A or B) and x ∈ (A or C)

x ∈ (A ∪ B) ∩ x ∈ (A ∩ C)

x ∈ (A ∪ B) ∩ (A ∪ C)

x∈ A ∪ (B ∩ C) => x ∈ (A ∪ B) ∩ (A ∪ C)

Therefore,

A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C).........(1)

Let x ∈ (A ∪ B) ∩ (A ∪ C). If x ∈ (A ∪ B) ∩ (A ∪ C) then x is in (A or B) and x is in (A or C).

x ∈ (A or B) and x ∈ (A or C)

{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}

x ∈ A or {x ∈ B and x ∈ C}

x ∈ A or {x ∈ (B and C)}

x ∈ A ∪ {x ∈ (B ∩ C)}

x ∈ A ∪ (B ∩ C)

x ∈ (A ∪ B) ∩ (A ∪ C) => x ∈ A ∪ (B ∩ C)

Therefore,

(A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C)..........(2)

So ,

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)