REPRESENTING THE GIVEN SETS IN VENN DIAGRAM

Understanding venn diagram with 2 circles :

Venn diagram is the method to represent the given data in circles.

Problem 1 :

Show A and B  on a Venn diagram if:

U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}

Solution:

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

A n B = {3, 4}

Problem 2 :

U = {4, 5, 6, 7, 8, 9, 10}, A = {6, 7, 9, 10}, B = {5, 6, 8, 9}

Solution:

A = {6, 7, 9, 10}, B = {5, 6, 8, 9}

AnB = {6, 9}

Problem 3 :

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

Solution:

A = {3, 5, 7, 9}, B = {4, 6, 8}

A n B = Null set

Problem 2 :

Suppose U = {x | x ≤ 30, x € Z⁺},

A = {prime numbers ≤ 30}, B = {multiple of 5 ≤ 30}

and C = {odd number ≤ 30}.

Use the Venn diagram shown to display the elements of the sets.

Solution:

U = {x | x ≤ 30, x € Z⁺}

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}

A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

, B = {5, 10, 15, 20, 25, 30}

C = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}

AnB = {5}

BnC = {15, 25}

CnA = {3, 5, 7, 11, 13, 17, 19, 23, 29}

AnBnC = {5}

Problem 3 :

Suppose U = {x | x ≤ 12, x € Z⁺} and A = {multiple of 3 ≤ 12}.

a. Show A on a Venn diagram

b. List the set A'

c. Find n(A')

d. True or false: If C = {1, 2, 4}, then C ⊆ A?

Solution:

U = {x | x ≤ 12, x € Z+} ---> (1)

A = {multiple of 3 ≤ 12} ---> (2)

U is a universal set and from (1)

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Since U is a universal set so A must be subset of U.

A = {3, 6, 9, 12} because (3, 6, 9, 12) are only positive multiple of 3 belongs to U.

a. Venn diagram

b.

A' = U - A 

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

Ignoring the elements of A from U and write the remaining elements.

A' = {1, 2, 4, 5, 7, 8, 10, 11}

c.

n(A') = n(U) - n(A)

= 12 - 4

n(A') = 8

d.

C = {1, 2, 4}

3 € A but 3 ∉ C

=> C ⊄ A

C = {1, 2, 4} then C ⊆ A is false.

Problem 4 :

Show this information on a Venn diagram:

U = {10, 11, 12, 13, 14, 15}, A = {10, 12, 14}, B = {11, 12, 13}

Solution:

A = {10, 12, 14}, B = {11, 12, 13}

A n B = {12}