Question

7. Verify De Morgan’s laws for set difference using the sets given below:
A = {1, 3, 5, 7, 9,11,13,15}, { B = 1, 2, 5, 7} and C = {3,9,10,12,13}

Answer 1

•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 = {1, 3, 5, 7, 9,11,13,15}, { B = 1, 2, 5, 7} and C = {3,9,10,12,13}

A\B = {3,9,11,13,15}
A\C = {1,5,7,11,15}
B U C = {1, 2, 3, 5, 7, 9, 10, 12, 13}
B ∩ C = { }

For, (A\B) ∩ (A\C) = {11,15} ………….(1)
& A\(B U C ) = {11,15}............(2)

From eq 1 & 2

(A\B) ∩ (A\C)= (A\B) ∩ (A\C)

For ,A\(B ∩ C )= {1, 3, 5, 7, 9, 11, 13, 15}.....................(3)
& (A\B) U (A\C) = {1, 3, 5, 7, 9, 11, 13, 15}....................(4)

From eq 3 & 4
A\(B ∩ C )= (A\B) U (A\C)

HOPE THIS WILL HELP YOU….

Answer 2

Hi ,

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De-morgan's Laws for set difference :

i ) A\(BUC) = ( A\B ) intersection ( A\C )

ii ) A\( B n C ) = ( A\B ) U ( A\C )

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It is given that ,

A = { 1,3,5,7,9,11,13,15 }

B = { 1,2,5,7 }

C = { 3,9,10,12,13 }

1 ) A \( B U C )

= A\( { 1,2,5,7 } U { 3,9,10,12,13 }

= A \ { 1,2,3,5,7,9,10,12,13 }

= { 1,3,5,7,9,11,13,15 } \{ 1,2,3,5,7,9,10,12,13 }

= { 11 ,15 } ----( 1 )

2 ) ( A\B ) n ( A \ C )

= ({ 1,3,5,7,9,11,13,15}\{1,2,5,7}) n ( {1,3,5,7,9,11,13,15}\{ 3,9,10,12,13 }

= { 3, 9, 11, 13 , 15 } n { 1 , 5 , 7 ,11 , 15 }

= { 11 , 15 } -----( 2 )

from ( 1 ) and ( 2 ) , we proved

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

Similarly ,

We can prove

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

I hope this helps you.

: )