DISTRIBUTIVE PROPERTY OF SET

Example 1 :

Given :

A = {0, 1, 2, 3, 4}

B = {1, -2, 3, 4, 5, 6}

C = {2, 4, 6, 7}

Show that

Au(BnC) = (AuB)n(AuC)

Also, verify the above using Venn diagram.

Solution :

BnC = {1, -2, 3, 4, 5, 6} n {2, 4, 6, 7}.

 BnC = {4, 6}

Au(BnC) = {0, 1, 2, 3, 4} u {4, 6}

  Au(BnC) = {0, 1, 2, 3, 4, 6} -----(1)

AuB = {0, 1, 2, 3, 4} u {1, - 2, 3, 4, 5, 6}

= {-2, 0, 1, 2, 3, 4, 5, 6}

AuC = {0, 1, 2, 3, 4} u {2, 4, 6, 7}

= {0, 1, 2, 3, 4, 6, 7}

(AuB)n(AuC) = {-2, 0, 1, 2, 3, 4, 5, 6} n {0, 1, 2, 3, 4, 6, 7}

(AuB)n(AuC) = {0, 1, 2, 3, 4, 6} ----(2)

From (1) and (2),

Au(BnC) = (AuB)n(AuC)

Venn Diagram :

Example 2 :

Given :

A = {x : - 3 ≤ x < 4, x ∊ R}

B = {x ; x < 5, x ∊ N}

C = {- 5, - 3, -1, 0, 1, 3}

Show that

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

Solution :

A = {x : - 3 ≤ x < 4, x ∊ R} ----> A = {-3, -2, -1, 0, 1, 2, 3}

B = {x ; x < 5, x ∊ N} ----> B = {1, 2, 3, 4}

C = {- 5, - 3, -1, 0, 1, 3}

B u C = {1, 2, 3, 4} u {- 5, - 3, - 1, 0, 1, 3}

  B u C = {-5, -3, -1, 0, 1, 2, 3, 4}

A n (B u C) = {-3, -2, -1, 0, 1, 2, 3} n {-5, -3, -1, 0, 1, 2, 3, 4}

A n (B u C) = {-3, -1, 0, 1, 2, 3} ----(1)

A n B = {-3, -2, -1, 0, 1, 2, 3} n {1, 2, 3, 4}

A n B = {1, 2, 3}

A n C = {-3, -2, -1, 0, 1, 2, 3} n {- 5, - 3, - 1, 0, 1, 3}

A n C = {-3, -1, 0, 1, 2, 3}

(A n B) u (A n C) = {1, 2, 3} U {-3, -1, 0, 1, 2, 3}

(A n B) u (A n C) = {-3, -1, 0, 1, 2, 3} ----(2)

From (1) and (2),

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

Example 3 :

Using the Venn diagram shown, list the elements of each of the following sets :

Solution :

a) A\B

Leave the common elements of A and B, choose the remaining elements of A.

A\B = {5}

b) A n B

Write the common elements of the sets A and B.

A n B = {8, 2}

c) (A n B)\C

= {8, 2} \ {2, 4, 3, 9}

Leave the common elements and remaining elements of A n B

= {8}

d)  AnBnC

Writing the common elements of the set A, B and C.

= {2}

e) AUBUC

Writing all elements in these three sets.

AUBUC = {2, 3, 4, 5, 6, 8, 9}

f)  (AUBUC)'

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

A U B U C = {2, 3, 4, 5, 6, 8, 9}

Leave the common elements in both sets and write the remaining elements U.

(AUBUC)' = {7}

Example 4 :

Let A = {k: k ∈ N, N ≤ 20}

B = {3k - 1, k ∈ N}

C = {2k + 1, k ∈ N}

Determing the sets

a) A n B n C

b)  (A n B) \ C

c) (A n C) \ B

Solution :

Let A = {k: k ∈ N, N ≤ 20}

N is a natural number starts with 1.

A= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

B = {3k - 1, k ∈ N}

• When k = 1
• 3k - 1 = 3(1) - 1 ==> 2
• When k = 2
• 3k - 1 = 3(2) - 1 ==> 5
• When k = 3
• 3k - 1 = 3(3) - 1 ==> 8
• When k = 4
• 3k - 1 = 3(4) - 1 ==> 11
• When k = 5
• 3k - 1 = 3(5) - 1 ==> 14
• When k = 6
• 3k - 1 = 3(6) - 1 ==> 17
• When k = 7
• 3k - 1 = 3(7) - 1 ==> 20

B = {2, 5, 8, 11, 14, 17, 20}

C = {2k + 1, k ∈ N}

• When k = 1
• 2k + 1 = 2(1) + 1 ==> 3
• When k = 2
• 2k + 1 = 2(2) + 1 ==> 5
• When k = 3
• 2k + 1 = 2(3) + 1 ==> 7
  
• When k = 4
• 2k + 1 = 2(4) + 1 ==> 9
• When k = 5
• 2k + 1 = 2(5) + 1 ==> 11
• When k = 6
• 2k + 1 = 2(6) + 1 ==> 13
• When k = 7
• 2k + 1 = 2(7) + 1 ==> 15
• When k = 8
• 2k + 1 = 2(8) + 1 ==> 17
• When k = 9
• 2k + 1 = 2(9) + 1 ==> 19

C = {3, 5, 7, 9, 11, 13, 15, 17, 19}

a) A n B n C

Common elements in all three sets A, B and C.

= {5, 11, 17}

b)  (A n B) \ C

(A n B) = {2, 5, 8, 11, 14, 17, 20}

(A n B) \ C = {2, 5, 8, 11, 14, 17, 20} \ {3, 5, 7, 9, 11, 13, 15, 17, 19}

= {2, 8, 14, 20}

c) (A n C) \ B

(A n C) = {3, 5, 7, 9, 11, 13, 15, 17, 19}

(A n C) \ B = {3, 5, 7, 9, 11, 13, 15, 17, 19} \ {2, 5, 8, 11, 14, 17, 20}

= {3, 7, 9, 13, 15, 19}

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