Given that [tex]$U = \{1,2, \ldots, 10\}$[/tex] is a universal set, [tex]$A = \{x: x \leq 5, x \in U[/tex] odd number, [tex]$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 U, A, B, and C in a Venn-diagram.
(iv) Find the percentage of [tex]$n _{ o }( A )$[/tex].
Answer (2)
Determine the elements of set A: A = { 1 , 3 , 5 } , and find its cardinality: n ( A ) = 3 .
Assume set B consists of even numbers in U: B = { 2 , 4 , 6 , 8 , 10 } , and identify set C as prime numbers in U: C = { 2 , 3 , 5 , 7 } .
Find the intersection of A, B, and C: A ∩ B ∩ C = ∅ , so n ( A ∩ B ∩ C ) = 0 .
Calculate the percentage of n ( A ) with respect to n ( U ) : 10 3 × 100 = 30% .
Explanation
Problem Analysis We are given the universal set U = { 1 , 2 , … , 10 } , the set A = { x : x ≤ 5 , x ∈ U , x is odd } and the set 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 U , A , B and C in a Venn diagram, and find the percentage of n ( A ) .
Finding Cardinality of A First, let's list the elements of set A. Since A contains odd numbers less than or equal to 5 that are in U, we have A = { 1 , 3 , 5 } . Therefore, the cardinality of A, denoted as n ( A ) , is the number of elements in A, which is n ( A ) = 3 .
Defining Set C and Assuming Set B Next, let's list the prime numbers in U to define set C. The prime numbers less than or equal to 10 are 2, 3, 5, and 7. So, C = { 2 , 3 , 5 , 7 } . The question mentions a set B, but it is not defined. Let's assume that B is the set of even numbers in U, so B = { 2 , 4 , 6 , 8 , 10 } .
Finding the Intersection of A, B, and C Now, let's find the intersection of A, B, and C: A ∩ B ∩ C . We have A = { 1 , 3 , 5 } , B = { 2 , 4 , 6 , 8 , 10 } , and C = { 2 , 3 , 5 , 7 } . The intersection A ∩ B ∩ C contains elements that are in all three sets. Comparing the sets, we see that there are no common elements, so A ∩ B ∩ C = ∅ . Thus, n ( A ∩ B ∩ C ) = 0 .
Venn Diagram Representation To represent the sets U, A, B, and C in a Venn diagram, U is the universal set, containing all numbers from 1 to 10. A contains 1, 3, and 5. B contains 2, 4, 6, 8, and 10. C contains 2, 3, 5, and 7. The sets A, B, and C are subsets of U. A and B are disjoint (they have no elements in common), while A and C have elements 3 and 5 in common, and B and C have element 2 in common.
Calculating the Percentage of n(A) Finally, let's calculate the percentage of n ( A ) with respect to the universal set U. We have n ( U ) = 10 , and n ( A ) = 3 . The percentage is ( n ( U ) n ( A ) ) × 100 = ( 10 3 ) × 100 = 30% .
Final Answer Therefore, the cardinality of set A is 3, the number of elements in A ∩ B ∩ C is 0, and the percentage of n ( A ) is 30%.
Examples
Understanding sets and their relationships is crucial in many real-world scenarios. For instance, in market research, you might define the universal set as all potential customers. Set A could be customers who prefer product A, set B those who prefer product B, and set C those who are influenced by a specific marketing campaign. Analyzing the intersections and cardinalities of these sets helps businesses tailor their strategies to target specific customer segments effectively, optimizing marketing efforts and product development.
The cardinality of set A is 3, the intersection A ∩ B ∩ C has 0 elements, and the percentage of n(A) is 30%.
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