ASSOCIATIVE PROPERTY OF SET WORKSHEET

Question 1 :

Given, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}.

Show that 

A u (B u C) = (A u B) u C

Also, verify the above using Venn diagram.

Question 2 :

Given A = {a, b, c, d}, B = {a, c, e} and C = {a, e}.

Show that

A n (B n C) = (A n B) n C

Also, verify the above using Venn diagram.

Question 3 :

Verify A u (B u C) = (A u B) u C for the following three sets A, B and C.

A = {x : x is a prime factor of 42}

B = {x | 5 < x ≤ 12, x ∊ N}

C = {1, 4, 5, 6}

Question 4 :

U = {x | 0 < x < 10, x ∈ N} 

A = {x | x is a prime number less than 10}

B = {x| x is even number less than 10}

C = {x|x is an odd natural number less than 10}

i) Represent sets U, A, B and C on Venn diagram.

ii) List the elements of A n C

iii)  List the elements of A n B.

iv)  Show that (A U B U C) = U

Question 5 :

U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 3, 4}

B = {3, 4, 5, 8}

C = {1, 3, 5, 7}

a)  Represent these sets in Venn diagram.

b)  Show that (A U B) U C = A U (B U C)

c)  Show that (A n B) n C = A n (B n C)

d) Show that (A\B)\C ≠ A \ (B\C)

1. Answer :

Given, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}.

BuC = {3, 4, 5, 6} u {5, 6, 7, 8}

Buc = {3, 4, 5, 6, 7, 8}

Au(BuC) = {1, 2, 3, 4, 5} u { 3, 4, 5, 6, 7, 8}

Au(BuC) = {1, 2, 3, 4, 5, 6, 7, 8} ----(1)

AuB = {1, 2, 3, 4, 5} u {3, 4, 5, 6}

AuB = {1, 2, 3, 4, 5, 6}

(AuB)uC = {1, 2, 3, 4, 5, 6} u {5, 6, 7, 8}

= {1, 2, 3, 4, 5, 6, 7, 8} ----(2)

From (1) and (2),

Au(BuC) = (AuB)uC

Venn Diagram :

2. Answer :

Given A = {a, b, c, d}, B = {a, c, e} and C = {a, e}.

BnC = {a, c, e} n {a, e}

BnC = {a, e}

An(BnC) = {a, b, c ,d} n {a, e}

An(BnC) = {a} ----(1)

AnB = {a, b, c, d} n {a, c, e}

AnB = {a, c}

(AnB)nC = {a, c} n {a,e}

(AnB)nC = {a} ----(2)

From (1) and (2),

An(BnC) = (AnB)nC

Venn Diagram :

3. Answer :

A = {x : x is a prime factor of 42} ---> A = {2, 3, 7}

B = {x | 5 < x ≤ 12, x ∊ N} ----> B = {6, 7, 8, 9, 10, 11, 12}

C = {1, 4, 5, 6}

BuC = {6, 7, 8, 9, 10, 11, 12} u {1, 4, 5, 6}

BuC = {1, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Au(BuC) = {2, 3, 7} u {1, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Au(BuC) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} ----(1)

AuB = {2, 3, 7} u {5, 6, 7, 8, 9, 10, 11, 12}

AuB = {2, 3, 5, 6, 7, 8, 9, 10, 11, 12}

(AuB)uC = {2, 3, 5, 6, 7, 8, 9, 10, 11, 12} u {1, 4, 5, 6}

(AuB)uC = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} ----(2)

From (1) and (2),

Au(BuC) = (AuB)uC

4. Answer :

U = {x | 0 < x < 10, x ∈ N} 

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {2, 3, 5, 7}

B = {2, 4, 6, 8}

C = {1, 3, 5, 7, 9}

i) Represent sets U, A, B and C on Venn diagram.

ii) A n C

= {2, 3, 5, 7} n {1, 3, 5, 7, 9}

A n C = {3, 5, 7}

iii)  List the elements of A n B.

= {2, 3, 5, 7} n {2, 4, 6, 8}

A n B = {2}

iv)  Show that (A U B U C) = U

(A U B U C) = {2, 3, 5, 7} U {2, 4, 6, 8} U {1, 3, 5, 7, 9}

= {1, 2, 3, 4, 5, 6, 7, 8, 9}

= The universal set.

Hence it is proved.

5. Answer :

U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 3, 4}

B = {3, 4, 5, 8}

C = {1, 3, 5, 7}

a)  Represent these sets in Venn diagram.

b)  (A U B) U C = A U (B U C)

(A U B) = {1, 2, 3, 4} U {3, 4, 5, 8}

= {1, 2, 3, 4, 5, 8}

(A U B) U C =  {1, 2, 3, 4, 5, 8} U {1, 3, 5, 7}

= {1, 2, 3, 4, 5, 7, 8} -----(1)

(B U C) = {3, 4, 5, 8} U  {1, 3, 5, 7}

= {3, 4, 5, 7, 8}

A U (B U C) =  {1, 2, 3, 4} U {3, 4, 5, 7, 8}

= {1, 2, 3, 4, 5, 7, 8}-----(2)

c) (A n B) n C = A n (B n C)

A n B = {1, 2, 3, 4} n {3, 4, 5, 8}

= {3, 4}

(A n B) n C = {3, 4} n  {1, 3, 5, 7}

= {3} ------(1)

(B n C) = {3, 4, 5, 8} n {1, 3, 5, 7}

= {3, 5}

A n (B n C) = {1, 2, 3, 4} n {3, 5}

= {3} ------(2)

d) (A\B)\C ≠ A \ (B\C)

A =

B = {3, 4, 5, 8}

C = {1, 3, 5, 7}

A\B = {1, 2, 3, 4}\{3, 4, 5, 8}

= {1, 2}

(A\B)\C = {1, 2}\{1, 3, 5, 7}

= {2} -----(1)

B\C = {3, 4, 5, 8}\ {1, 3, 5, 7}

= {4, 8}

A\(B\C) = {1, 2, 3, 4}\{4, 8}

= {1, 2, 3} -----(2)

(1) and (2) are not equal.

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