Question

8. Let A = {10,15, 20, 25, 30, 35, 40, 45, 50} , B = {1,5,10,15, 20, 30}
and C = {7, 8,15,20, 35,45, 48}. Verify A\(B u C) = (A\ B) U (A\ C) .

Answer 1

QUESTION :
Let A = {10,15, 20, 25, 30, 35, 40, 45, 50} , B = {1,5,10,15, 20, 30}
and C = {7, 8,15,20, 35,45, 48}.
Verify A\(B ∩ C ) = (A\B) U (A\C)

•A\B means removing all elements of B from A.
Union of two sets :
The union of the sets A and B is the set of all the element that belongs to either A or B or both. It is denoted by A U B(“A union B”).
Intersection of two sets :
The intersection of the sets a and b is the set of all the elements which belong to both A and B. It is denoted by A ∩ B (“ A intersection B”).
•If A and B do not have any element in common then A ∩ B= a null set = Ø

•A’ = The complementary set of A
• To find A’ , list all the members of the universal set U which are not members of A.
A’ = U - A
De-morgan's Laws for set Difference
A\(B U C ) = (A\B) ∩ (A\C)
A\(B ∩ C ) = (A\B) U (A\C)

SOLUTION :
GIVEN :
A = {10,15, 20, 25, 30, 35, 40, 45, 50} , B = {1,5,10,15, 20, 30}
C = {7, 8,15,20, 35,45, 48}

We have to verify :
A\(B ∩ C ) = (A\B) U (A\C)

L.H.S
B ∩ C = {15,20 }
A\(B U C ) = {10, 25, 30, 35, 40, 45, 50}

R.H.S
A\B = {25, 35, 40, 45, 50}
A\C = {10, 25, 30, 40, 50}

(A\B) U (A\C) = {10, 25, 30, 35, 40, 45, 50}

L.H.S = R.H.S

HOPE THIS WILL HELP YOU...

Answer 2

Hi ,

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We know that ,

A U B = { x/x€ A or x € B }

A\B = { x/ x € A and x doesn't belongs to x }

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A = { 10,15,20,25,30,35,40,45,50 }

B = { 1, 5, 10, 15, 20 , 30 }

C = { 7 ,8, 15 ,20 ,35 ,45 , 48 }

LHS = A\( B U C )

= A \( { 1,5,10,15,20,30} U { 7,8,15,20,35,45,48})

= A \{ 1,5,7,8,10,15,20,30,35,45,48 }

= { 10,15,20,25,30,35,40,45,50}\{1,5,7,8,10,15,20,30,35,45,48 }

= { 25 , 40 , 50 } ------( 1 )

RHS = ( A\B ) U ( A\C )

= ({10,15,20,25,30,35,40,45,50}\{1,5,10,15,20,30})

U ({10,15,20,25,30,35,40,45,50}\{7,8,15,20,35,45,48 } )

= { 25,35, 40,45,50} U { 10,25,30,40,50 }

= { 10, 25,30, 35, 40,45, 50 }----( 2 )

from ( 1 ) and ( 2 ) , we conclude that ,

LHS ≠ RHS

A \( B U C ) ≠ ( A\B ) U ( A\C )

I hope this helps you.

: )