Q. 64.0( 6 Votes )

If A = {1, 3, 5)

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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