(iv) Find the percentage of [tex]$n _{ o }( A )$[/tex].
b) [tex]$U={x: x$[/tex] is a natural number, [tex]$x \leq 20}, A={y: y$[/tex] is a factor of 18[tex]$}, B={z: z$[/tex] is a multiple of [tex]$3, z \in U}$[/tex] and [tex]$C={w: w$[/tex] is a perfect square number, [tex]$w \in U}$[/tex]}.
(i) What is the value of cardinality [tex]$n ( C )$[/tex]?
(ii) How many elements are there in [tex]$A \cap C$[/tex]?
(iii) Show the relation of the sets [tex]$U , A , B$[/tex] and C in a Venn-diagram and prove that [tex]$n ( A \cup B \cup C )= n ( A \cup B )+ n _{ o }( C )$[/tex].
(iv) Shilpa said that [tex]$n(A \cup B)+n_o(C)$[/tex] and [tex]$n_o(A \cup B)+n(C)$[/tex] are equal. Analyze her statement.

Determine the elements of sets U , A , B , and C , and calculate their cardinalities.
Find the intersections of the sets: A ∩ B , A ∩ C , B ∩ C , and A ∩ B ∩ C , and calculate their cardinalities.
Calculate n o ​ ( A ) using the formula n o ​ ( A ) = n ( A ) − n ( A ∩ B ) − n ( A ∩ C ) + n ( A ∩ B ∩ C ) , and find the percentage of n o ​ ( A ) with respect to n ( U ) .
Verify the equation n ( A ∪ B ∪ C ) = n ( A ∪ B ) + n o ​ ( C ) and analyze Shilpa's statement about the equality of n ( A ∪ B ) + n o ​ ( C ) and n o ​ ( A ∪ B ) + n ( C ) .

The percentage of n o ​ ( A ) is 5% ​ , n ( C ) = 4 ​ , n ( A ∩ C ) = 2 ​ , and Shilpa's statement is correc t ​ .
Explanation

Identifying the Sets First, let's identify the elements in each set:

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

Cardinality of the Sets Now, let's find the number of elements in each set:

n ( U ) = 20 n ( A ) = 6 n ( B ) = 6 n ( C ) = 4

Intersections of the Sets Next, we need to find the intersections of the sets:

A ∩ B = { 3 , 6 , 9 , 18 } A ∩ C = { 1 , 9 } B ∩ C = { 9 } A ∩ B ∩ C = { 9 }

Cardinality of Intersections Now, let's find the number of elements in each intersection:

n ( A ∩ B ) = 4 n ( A ∩ C ) = 2 n ( B ∩ C ) = 1 n ( A ∩ B ∩ C ) = 1

Calculating n_o(A) To find n o ​ ( A ) , which represents the number of elements that belong to A only, we use the formula:

n o ​ ( A ) = n ( A ) − n ( A ∩ B ) − n ( A ∩ C ) + n ( A ∩ B ∩ C ) n o ​ ( A ) = 6 − 4 − 2 + 1 = 1

Percentage of n_o(A) Now, we calculate the percentage of n o ​ ( A ) with respect to the universal set U :

n ( U ) n o ​ ( A ) ​ × 100 = 20 1 ​ × 100 = 5%

Cardinality of C (i) The cardinality of set C is n ( C ) = 4 .

Elements in A ∩ C (ii) The number of elements in A ∩ C is n ( A ∩ C ) = 2 .

Venn Diagram and Verification (iii) The relation of the sets U , A , B , and C can be represented in a Venn diagram. We can verify that n ( A ∪ B ∪ C ) = n ( A ∪ B ) + n o ​ ( C ) .


First, let's find A ∪ B = { 1 , 2 , 3 , 6 , 9 , 12 , 15 , 18 } . So, n ( A ∪ B ) = 8 .
Next, A ∪ B ∪ C = { 1 , 2 , 3 , 4 , 6 , 9 , 12 , 15 , 16 , 18 } . So, n ( A ∪ B ∪ C ) = 10 .
Now, let's find n o ​ ( C ) = n ( C ) − n ( A ∩ C ) − n ( B ∩ C ) + n ( A ∩ B ∩ C ) = 4 − 2 − 1 + 1 = 2 .
Finally, we check if n ( A ∪ B ∪ C ) = n ( A ∪ B ) + n o ​ ( C ) : 10 = 8 + 2 , which is true.

Analyzing Shilpa's Statement (iv) Shilpa said that n ( A ∪ B ) + n o ​ ( C ) and n o ​ ( A ∪ B ) + n ( C ) are equal. Let's analyze her statement.

We already know that n ( A ∪ B ) = 8 and n o ​ ( C ) = 2 . So, n ( A ∪ B ) + n o ​ ( C ) = 8 + 2 = 10 .
Now, we need to find n o ​ ( A ∪ B ) . A ∪ B = { 1 , 2 , 3 , 6 , 9 , 12 , 15 , 18 } .
C ∩ ( A ∪ B ) = { 1 , 9 } . So, n ( C ∩ ( A ∪ B )) = 2 .
n o ​ ( A ∪ B ) = n ( A ∪ B ) − n ( C ∩ ( A ∪ B )) = 8 − 2 = 6 .
Now, let's calculate n o ​ ( A ∪ B ) + n ( C ) = 6 + 4 = 10 .
Since n ( A ∪ B ) + n o ​ ( C ) = 10 and n o ​ ( A ∪ B ) + n ( C ) = 10 , Shilpa's statement is correct.

Final Answer Therefore: (iv) The percentage of n o ​ ( A ) is 5% .
(i) n ( C ) = 4 .
(ii) n ( A ∩ C ) = 2 .
(iii) n ( A ∪ B ∪ C ) = n ( A ∪ B ) + n o ​ ( C ) is proven. (iv) Shilpa's statement is correct.

Examples
Understanding sets and their relationships is crucial in various real-life scenarios. For instance, in market research, companies use Venn diagrams to analyze customer preferences. If set A represents customers who prefer product X, set B represents those who prefer product Y, and set C represents those who prefer product Z, the intersections and unions of these sets help companies tailor their marketing strategies. By understanding the number of customers who prefer only one product versus those who prefer multiple products, companies can optimize their product offerings and advertising campaigns to maximize customer satisfaction and sales. This approach ensures resources are allocated effectively, targeting the right customer segments with the right products and promotions.

Answered by GinnyAnswer | 2025-07-05