Question

4. By taking suitable sets A, B, C, verify the following results:

(ii) A × (B ∪ C)=(A × B) ∪ (A × C).

Answer 1

Cartesian Product :
M x N is the set containing all the possible ordered pair (m ,n ) : m £ M and n £ N.
where £ means belongs to.

Steps :
1) Let A = { 1,2}
B ={ 3 }
C = { 4 }

Then,
LHS = A x ( B U C)
= { 1,2} x( { 3 } U {4} )
= { 1,2} x { 3,4 }
= (1,3 ) ,(1,4) , (2,3) , (2,4)

RHS = ( A x B) U ( A x C)
= ( { 1,2} x { 3} ) U ( {1,2} U {4} )
= (1,3) , (2,3) , ( 1,4) , (2,4)

Since, LHS = RHS
Hence Proved.

Answer 2

Solution :

Let A = { 2,3 },

B = { 1,3},

And

C = { 5,7 }

i ) LHS = A × ( BUC )

= A × ({1,3} U { 5,7 })

= A×{1,3,5,7}

= {2,3} ×{ 1,3,5,7 }

={(2,1),(2,3),(2,5),(2,7),(3,1),(3,3),

(3,5),(3,7)} ---( 1 )

ii ) RHS = (A×B) U(A×C)

=({2,3}×{1,3})U({2,3}×{5,7})

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

{ (2,5),(2,7),(3,5),(3,7) }

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

(2,7),(3,5),(3,7) }

={ (2,1),(2,3),(2,5),(2,7),

(3,1),(3,3),(3,5),3,7) } ---( 2 )

From (1 ) & ( 2 ) , we conclude that

LHS = RHS

A × ( BUC ) = ( A×B ) U ( A×C )

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