Answer :

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

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