(i) Given: A = {1, 3, 5}, B = {3, 4} and C = {2, 3}

L. H. S = A × (B ⋃ C) 

By the definition of the union of two sets, (B ⋃ C) = {2, 3, 4} 

= {1, 3, 5} × {2, 3, 4} 

Now, by the definition of the Cartesian product, 

Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P and Q, .i.e. 

P × Q = {(p, q) : p Є P, q Є Q} 

= {(1, 2), (1, 3), (1, 4), (3, 2), (3, 3), (3, 4), (5, 2), (5, 3), (5, 4)} 

R. H. S = (A × B) ⋃ (A × C) 

Now, A × B = {1, 3, 5} × {3, 4} 

= {(1, 3), (1, 4), (3, 3), (3, 4), (5, 3), (5, 4)} 

and A × C = {1, 3, 5} × {2, 3} 

= {(1, 2), (1, 3), (3, 2), (3, 3), (5, 2), (5, 3)} 

Now, we have to find (A × B) ⋃ (A × C) 

So, by the definition of the union of two sets, 

(A × B) ⋃ (A × C) = {(1, 2), (1, 3), (1, 4), (3, 2), (3, 3), (3, 4), (5, 2), (5, 3), (5, 4)} 

= L. H. S 

∴ L. H. S = R. H. S is verified 

(ii) Given: A = {1, 3, 5}, B = {3, 4} and C = {2, 3} 

L. H. S = A × (B ⋂ C) 

By the definition of the intersection of two sets, (B ⋂ C) = {3} 

= {1, 3, 5} × {3} 

Now, by the definition of the Cartesian product, 

Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P and Q, .i.e. 

P × Q = {(p, q) : p Є P, q Є Q}

= {(1, 3), (3, 3), (5, 3)}

R. H. S = (A × B) ⋂ (A × C) 

Now, A × B = {1, 3, 5} × {3, 4}

= {(1, 3), (1, 4), (3, 3), (3, 4), (5, 3), (5, 4)} 

and A × C = {1, 3, 5} × {2, 3} 

= {(1, 2), (1, 3), (3, 2), (3, 3), (5, 2), (5, 3)} 

Now, we have to find (A × B) ⋂ (A × C) 

So, by the definition of the intersection of two sets, 

(A × B) ⋂ (A × C) = {(1, 3), (3, 3), (5, 3)} 

= L. H. S 

∴ L. H. S = R. H. S is verified