Question

If A = {1, 4}; B = {4, 5}; C = {5, 7}, verify that
A×(B ∩C) = (A×B)∩(A×C)

Answer 1

Step-by-step explanation:

Given :-

A = {1, 4};

B = {4, 5};

C = {5, 7}

To find :-

Verify that A×(B ∩C) = (A×B)∩(A×C) .

Solution :-

Given sets are :

A = {1, 4};

B = {4, 5};

C = {5, 7}

Finding A×(B ∩C) :-

To find A×(B ∩C) , first find (B ∩C) and then find

A×(B ∩C)

(B ∩C)

=> { 4,5} ∩ { 5,7}

= {5}

Now,

A×(B ∩C)

=> {1,4} × {5}

=> { (1,4) ,(1,5) }

A×(B ∩C) = { (1,4) ,(1,5) } ---------(1)

Finding (A×B)∩(A×C):-

To find (A×B)∩(A×C) ,first find A×B and A×C and then after find (A×B)∩(A×C).

A×B

=> {1,4}×{4,5}

=> { (1,4),(1,5),(4,4),(4,5)}

and

A×C

=> {1,4}×{5,7}

=> {(1,5),(1,7),(4,5),(4,7)}

now

(A×B)∩(A×C)

=> { (1,4),(1,5),(4,4),(4,5)} ∩{(1,5),(1,7),(4,5),(4,7)}

=> (A×B)∩(A×C) = {(1,5),(4,5)}----------(2)

From (1)&(2)

A×(B ∩C) = (A×B)∩(A×C) is verified.

Answer :-

A×(B ∩C) = (A×B)∩(A×C) for any non-empty sets A,B and C

Used formulae:-

Let A,B and C be any non empty sets then

• The set of all common elements in both A and B sets is called the intersection of A and B and it is denoted by A∩B .

• The set of all order pairs such that first element belongs to A and the second element belongs to B is called Cartesian Product of A and B and it is denoted by A×B .

• A×B ={(x,y),x€A,x€B}

• A∩B = { x/ x€ A and x€ B }