Question

If a b and c are three set than prove that a x (buc)=(axb) u (axc)

Answer 1

A × (B U C) = (A × B) U (A × C)

Proof:

Let, (x, y) ∈ A × (B U C)

⇒ x ∈ A and y ∈ (B U C)

⇒ x ∈ A and (y ∈ B or y ∈ C)

⇒ (x ∈ A and y ∈ B) or (x ∈ A and y ∈ C)

⇒ (x, y) ∈ (A × B) or (x, y) ∈ (A × C)

⇒ (x, y) ∈ (A × B) U (A × C)

∴ A × (B U C) ⊆ (A × B) U (A × C) ..... (1)

Again let, (x, y) ∈ (A × B) U (A × C)

⇒ (x, y) ∈ (A × B) or (x, y) ∈ (A × C)

⇒ (x ∈ A and y ∈ B) or (x ∈ A and y ∈ C)

⇒ x ∈ A and (y ∈ B or y ∈ C)

⇒ x ∈ A and y ∈ (B U C)

⇒ (x, y) ∈ A × (B U C)

∴ (A × B) U (A × C) ⊆ A × (B U C) ..... (2)

From (1) and (2), we get

A × (B U C) = (A × B) U (A × C)

Thus proved.

Answer 2

A × (B U C) = (A × B) U (A × C)

Proof:

Let, (x, y) ∈ A × (B U C)

⇒ x ∈ A and y ∈ (B U C)

⇒ x ∈ A and (y ∈ B or y ∈ C)

⇒ (x ∈ A and y ∈ B) or (x ∈ A and y ∈ C)

⇒ (x, y) ∈ (A × B) or (x, y) ∈ (A × C)

⇒ (x, y) ∈ (A × B) U (A × C)

∴ A × (B U C) ⊆ (A × B) U (A × C) ..... (1)

Again let, (x, y) ∈ (A × B) U (A × C)

⇒ (x, y) ∈ (A × B) or (x, y) ∈ (A × C)

⇒ (x ∈ A and y ∈ B) or (x ∈ A and y ∈ C)

⇒ x ∈ A and (y ∈ B or y ∈ C)

⇒ x ∈ A and y ∈ (B U C)

⇒ (x, y) ∈ A × (B U C)

∴ (A × B) U (A × C) ⊆ A × (B U C) ..... (2)

From (1) and (2), we get

A × (B U C) = (A × B) U (A × C)

Thus proved.

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