(i) Given: 

A = {x ϵ W : x < 2} 

Here, W denotes the set of whole numbers (non – negative integers). 

∴ A = {0, 1} 

[∵ It is given that x < 2 and the whole numbers which are less than 2 are 0 & 1] 

B = {x ϵ N : 1 < x ≤ 4} 

Here, N denotes the set of natural numbers. 

∴ B = {2, 3, 4} 

[∵ It is given that the value of x is greater than 1 and less than or equal to 4] and C = {3, 5} 

L. H. S = A × (B ⋃ C) 

By the definition of the union of two sets, (B ⋃ C) = {2, 3, 4, 5} 

= {0, 1} × {2, 3, 4, 5} 

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}

= {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 5)} 

R. H. S = (A × B) ⋃ (A × C) 

Now, A × B = {0, 1} × {2, 3, 4} 

= {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)} 

and A × C = {0, 1} × {3, 5} 

= {(0, 3), (0, 5), (1, 3), (1, 5)} 

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

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

(A × B) ⋃ (A × C) = {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 5)} 

= L. H. S 

∴ L. H. S = R. H. S is verified

(ii) Given: 

A = {x ϵ W : x < 2} 

Here, W denotes the set of whole numbers (non – negative integers). 

∴ A = {0, 1} 

[∵ It is given that x < 2 and the whole numbers which are less than 2 are 0, 1] 

B = {x ϵ N : 1 < x ≤ 4} 

Here, N denotes the set of natural numbers. 

∴ B = {2, 3, 4} 

[∵ It is given that the value of x is greater than 1 and less than or equal to 4] and C = {3, 5} 

L. H. S = A × (B ⋂ C) 

By the definition of the intersection of two sets, (B ⋂ C) 

= {3} = {0, 1} × {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}

= {(0, 3), (1, 3)} 

R. H. S = (A × B) ⋂ (A × C) 

Now, A × B = {0, 1} × {2, 3, 4} 

= {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)} 

and A × C = {0, 1} × {3, 5} 

= {(0, 3), (0, 5), (1, 3), (1, 5)} 

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

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

(A × B) ⋂ (A × C) = {(0, 3), (1, 3)} 

= L. H. S 

∴ L. H. S = R. H. S is verified