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