 # If A = {1, 2, 3}, B = {4}, C = {5}, then verify that: Question:

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:

(i) A × (B ∪ C) = (A × B) ∪ (A × C)

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

(iii) A × (B − C) = (A × B) − (A × C)

Solution:

Given:

A = {1, 2, 3}, B = {4} and C = {5}

(i) A × (B ∪ C) = (A × B) ∪ (A × C)

We have:

(B ∪ C) = {4, 5}

LHS: A × (B ∪ C)  = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

Now,

(A × B) = {(1, 4), (2, 4), (3, 4)}

And,

(A × C) = {(1, 5), (2, 5), (3, 5)}

RHS: (A × B) ∪ (A × C) = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)}

∴ LHS = RHS

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

We have:

$(B \cap C)=\phi$

$\mathrm{LHS}: A \times(B \cap C)=\phi$

And,

(A × B) = {(1, 4), (2, 4), (3, 4)}

(A × C) = {(1, 5), (2, 5), (3, 5)}

$\mathrm{RHS}:(A \times B) \cap(A \times C)=\phi$

∴ LHS = RHS

(iii) A × (B − C) = (A × B) − (A × C)

We have:

$(B-C)=\phi$

$\mathrm{LHS}: A \times(B-C)=\phi$

Now,

(A × B) = {(1, 4), (2, 4), (3, 4)}

And,

(A × C) = {(1, 5), (2, 5), (3, 5)}

RHS: $(A \times B)-(A \times C)=\phi$

∴ LHS = RHS