Question

11. For A ={-3,-1,0,4,6,8,10}, { B =-1,-2, 3,4,5,6} and C = {-1, 2,3,4,5,7},
show that (i) Au(BuC)=(AUB)u(AUC) (ii) Au(BUC )=(AuB)U(AuC) (iii) Verify (i) using Venn diagram (iv) Verify (ii) using Venn diagram.

Answer 1

Question :
For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that
(i) A U (B ∩ C) = (A U B) ∩ (A U C)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)
(iii) Verify using Venn diagrams


Venn diagram :
A Venn diagram is a closed figure used to denote a set. Points within such a closed figure represent the elements of the set. Venn diagrams can be used to represent the relationship between sets. That Universal set U is usually represented by a rectangle.

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


SOLUTION :
GIVEN :
A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7}

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

L.H.S
A U (B ∩ C)

(B ∩ C) = {-1,-2,3,4,5,6} ∩ {-1,2,3,4,5,7}
= {-1,3,4,5}

A U (B ∩ C) = {-3,-1,0,4,6,8,10} U {-1,3,4,5}

A U (B ∩ C) = {-3,-1,0,3,4,5,6,8,10}

R.H.S
(A U B) ∩ (A U C)

(A U B) = {-3,-1,0,4,6,8,10} U {-1,-2,3,4,5,6}
= {-3,-2,-1,0,3,4,5,6,8,10}

(A U C) = {-3,-1,0,4,6,8,10} U {-1,2,3,4,5,7}
= {-3,-1,0,2,3,4,5,6,7,8,10}

(AUB)∩(AUC)={-3,-2,-1,0,3,4,5,6,8,10} ∩ {-3,-1,0,2,3,4,5,6,7,8,10}

(AUB)∩(AUC) = {-3,-1,0,3,4,5,6,8,10}

A U (B ∩ C) = (A U B) ∩ (A U C)

L.H.S = R.H.S
Hence proved

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

GIVEN :
A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7}

L.H.S
(B U C) = {-1,-2,3,4,5,6} U {-1,2,3,4,5,7}
= {-2,-1,2,3,4,5,6,7}

A ∩ (B U C) = {-3,-1,0,4,6,8,10} ∩ {-2,-1,2,3,4,5,6,7}

A ∩ (B U C) = {-1,4,6}

R.H.S
(A ∩ B) = {-3,-1,0,4,6,8,10} ∩ {-1,-2,3,4,5,6}
= {-1,4,6}

(A ∩ C) = {-3,-1,0,4,6,8,10} ∩ {-1,2,3,4,5,7}
= {-1,4}

(A ∩ B) U (A ∩ C) = {-1,4,6} U {-1,4}

(A ∩ B) U (A ∩ C) = {-1,4,6}

A ∩ (B U C) = (A ∩ B) U (A ∩ C)
L.H.S = R.H.S

iii) Venn diagram is in the attachment.

HOPE THIS WILL HELP YOU…

Answer 2

we know that ,

If A ,B two sets ,

A U B = { x/ x€A or x € B }

A n B ={ x/x € A and x € B}

AU( BUC ) = ( A U B ) U C

Associative proporty .

I hope this helps you.

: )