5. a) Given that [tex]$U =\{1,2, \ldots, 10\}$[/tex] is a universal set. [tex]$A =\{ x : x \leq 5, x \in U \}, B =\{ y : y$[/tex] odd number, [tex]$y \in U \}$[/tex] and [tex]$C =\{ z : z$[/tex] is a prime number, [tex]$z \in U \}$[/tex].
(i) What is the cardinality of set A?
(ii) How many elements are there in [tex]$A \cap B \cap C$[/tex]?
(iii) Show the relation of the sets [tex]$U , A , B$[/tex] and C in a Venn-diagram and prove [tex]$n(A \cup B \cup C)=n(A)+n(B)+n(C)-n(A \cap B)-n(B \cap C)-n(C \cap A)+n(A \cap B \cap C)$[/tex]
(iv) Find the percentage of [tex]$n _0(A)$[/tex].

The cardinality of set A is 5, with 2 elements in the intersection of sets A, B, and C. The inclusion-exclusion principle results in a union count of 8, and the percentage of elements unique to set A is 10%.
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Answered by Anonymous | 2025-07-05

Determine the cardinality of set A: Since A = { 1 , 2 , 3 , 4 , 5 } , then n ( A ) = 5 .
Find the intersection A ∩ B ∩ C : Given A = { 1 , 2 , 3 , 4 , 5 } , B = { 1 , 3 , 5 , 7 , 9 } , and C = { 2 , 3 , 5 , 7 } , the intersection is A ∩ B ∩ C = { 3 , 5 } , so n ( A ∩ B ∩ C ) = 2 .
Verify the inclusion-exclusion principle: n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) − n ( A ∩ B ) − n ( B ∩ C ) − n ( C ∩ A ) + n ( A ∩ B ∩ C ) = 5 + 5 + 4 − 3 − 3 − 3 + 2 = 7 .
Calculate the percentage of n 0 ​ ( A ) : A ∖ ( B ∪ C ) = { 4 } , so the percentage is 10 1 ​ × 100% = 10% ​ .

Explanation

Problem Analysis We are given the universal set U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } , and three subsets: A = { x : x ≤ 5 , x ∈ U } , B = { y : y is an odd number , y ∈ U } , and C = { z : z is a prime number , z ∈ U } . We need to find the cardinality of set A, the number of elements in A ∩ B ∩ C , show the relation of the sets in a Venn-diagram and prove the inclusion-exclusion principle, and find the percentage of n 0 ​ ( A ) .

Cardinality of Set A (i) The set A contains all elements in U that are less than or equal to 5. Thus, A = { 1 , 2 , 3 , 4 , 5 } . The cardinality of A, denoted as n ( A ) , is the number of elements in A. Therefore, n ( A ) = 5 .

Intersection of A, B, and C (ii) We need to find the intersection of A, B, and C, which means we need to find the elements that are common to all three sets. We have: A = { 1 , 2 , 3 , 4 , 5 } B = { 1 , 3 , 5 , 7 , 9 } C = { 2 , 3 , 5 , 7 } The intersection A ∩ B ∩ C is the set of elements that are in all three sets. Comparing the sets, we find that the common elements are 3 and 5. Thus, A ∩ B ∩ C = { 3 , 5 } . The number of elements in A ∩ B ∩ C is n ( A ∩ B ∩ C ) = 2 .

Venn Diagram and Inclusion-Exclusion Principle (iii) To show the relation of the sets in a Venn-diagram, we draw a rectangle representing the universal set U, and three overlapping circles representing the sets A, B, and C. The overlapping regions represent the intersections of the sets. To prove the inclusion-exclusion principle, we need to calculate the number of elements in A ∪ B ∪ C and verify the formula: n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) − n ( A ∩ B ) − n ( B ∩ C ) − n ( C ∩ A ) + n ( A ∩ B ∩ C ) First, let's find A ∪ B ∪ C . This set contains all elements that are in A, B, or C (or any combination of them). A ∪ B ∪ C = { 1 , 2 , 3 , 4 , 5 , 7 , 9 } . Thus, n ( A ∪ B ∪ C ) = 7 .
Now, let's calculate the individual terms: n ( A ) = 5 n ( B ) = 5 n ( C ) = 4 A ∩ B = { 1 , 3 , 5 } , so n ( A ∩ B ) = 3 B ∩ C = { 3 , 5 , 7 } , so n ( B ∩ C ) = 3 C ∩ A = { 2 , 3 , 5 } , so n ( C ∩ A ) = 3 A ∩ B ∩ C = { 3 , 5 } , so n ( A ∩ B ∩ C ) = 2 Now, plug these values into the formula: n ( A ∪ B ∪ C ) = 5 + 5 + 4 − 3 − 3 − 3 + 2 = 14 − 9 + 2 = 5 + 2 = 7 Since n ( A ∪ B ∪ C ) = 7 , the formula is verified.

Percentage of n_0(A) (iv) We need to find the percentage of n 0 ​ ( A ) , which represents the number of elements that belong to A only. This means we want to find the elements in A that are not in B or C. In other words, we want to find A ∖ ( B ∪ C ) .
First, let's find B ∪ C = { 1 , 2 , 3 , 5 , 7 , 9 } .
Now, A ∖ ( B ∪ C ) is the set of elements in A that are not in B ∪ C . Comparing the sets, we have: A = { 1 , 2 , 3 , 4 , 5 } B ∪ C = { 1 , 2 , 3 , 5 , 7 , 9 } A ∖ ( B ∪ C ) = { 4 } So, n ( A ∖ ( B ∪ C )) = 1 .
To find the percentage of n 0 ​ ( A ) with respect to the universal set U, we calculate: n ( U ) n ( A ∖ ( B ∪ C )) ​ × 100% = 10 1 ​ × 100% = 10%

Final Answer (i) The cardinality of set A is 5. (ii) The number of elements in A ∩ B ∩ C is 2. (iii) The inclusion-exclusion principle is verified. (iv) The percentage of n 0 ​ ( A ) is 10%.


Examples
Understanding set theory is crucial in many real-world applications, such as database management and data analysis. For instance, imagine you're organizing a survey to understand customer preferences for different product features. Set A could represent customers who like feature A, set B those who like feature B, and set C those who like feature C. By analyzing the intersections and unions of these sets, you can identify which features are most popular among different customer segments, allowing you to make informed decisions about product development and marketing strategies. The inclusion-exclusion principle helps you accurately count the total number of customers who like at least one feature, avoiding double-counting those who like multiple features. This ensures your analysis is precise and reliable, leading to better business outcomes.

Answered by GinnyAnswer | 2025-07-05