Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that (i) A × (B ∩ C) = (A × B) ∩ (A × C) (ii) A × C is a subset of B × D

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Solution

(i)

The sets A, B, C and D are given { 1,2 }, { 1,2,3,4 }, { 5,6 } and { 5,6,7,8 }respectively.

The Cartesian product of sets A and ( B∩C ) is,

B∩C={ 1,2,3,4 }∩{ 5,6 } =ϕ

Here, ϕ is a null vector.

Then,

A×( B∩C )={ 1,2 }×ϕ =ϕ (1)

The Cartesian product of A×B is,

A×B={ ( 1,1 ),( 1,2 ),( 1,3 ),( 1,4 ),( 2,1 ),( 2,2 ),( 2,3 ),( 2,4 ) }

The Cartesian product of A×C is,

A×C={ ( 1,5 ),( 1,6 ),( 2,5 ),( 2,6 ) }

Then, the value of ( A×B )∩( A×C ) is,

( A×B )∩( A×C )=ϕ(2)

Hence, equations (1) and (2) are the same, so the system is verified.

(ii)

The value of Cartesian product of A×C is,

A×C={ ( 1,5 ),( 1,6 ),( 2,5 ),( 2,6 ) }(3)

The value of Cartesian product of B×D is,

B×D={ ( 1,5 ),( 1,6 ),( 1,7 ),( 1,8 ),( 2,5 ),( 2,6 ),( 2,7 ),( 2,8 ), ( 3,5 ),( 3,6 ),( 3,7 ),( 3,8 ),( 4,5 ),( 4,6 ),( 4,7 ),( 4,8 ) } (4)

All the elements in the set A×C are the elements of B×D.

Hence, A×C is a subset of B×D.

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Similar questions

A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}

D = {5, 6, 7, 8}

A \times (B \cap C) = (A \times B) \cap (A \times C)

A \times C

B \times D

A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}

D = {5, 6, 7, 8} .

(i) A \times (B \cap C) = (A \times B) \cap (A \times C)

(i i) A \times C

B \times D

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that :

(i) $A \times C \subset B \times D$

(ii) $A \times (B \cap C) = (A \times B) \cap (A \times C)$

A = {1, 2}

B = {1, 2, 3, 4}

C = {5, 6}

D = {5, 6, 7, 8}

A \times (B \cap C) = (A \times B) \cap (A \times C)

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