Consider the sets:
[tex]$U =\{x \mid x$[/tex] is a real number [tex]$\}$[/tex]
[tex]$A=\{x \mid x$[/tex] is an odd integer [tex]$\}$[/tex]
[tex]$R =\{x \mid x=3,7,11,27\}$[/tex]
Is [tex]$R \subset A$[/tex]?
A. yes, because all the elements of set R are in set A
B. no, because each element in set R is not represented in set A
C. yes, because all the elements of set A are in set R
D. no, because each element in set A is not represented in set R
Answer (1)
Define sets U , A , and R .
Verify that each element in R = { 3 , 7 , 11 , 27 } is an odd integer.
Confirm that all elements of R are in A , where A is the set of all odd integers.
Conclude that R ⊂ A , so the answer is yes, because all the elements of set R are in set A.
Explanation
Understanding the Problem We are given three sets: U , A , and R . Our goal is to determine if R is a subset of A , denoted as R ⊂ A . This means we need to check if every element in set R is also an element in set A .
Analyzing Set U Set U is the set of all real numbers. This information is not directly relevant to determining if R ⊂ A , but it defines the universal set from which the other sets are derived.
Analyzing Set A Set A is defined as the set of all odd integers. An integer is odd if it can be written in the form 2 k + 1 , where k is an integer.
Analyzing Set R Set R is given as R = { 3 , 7 , 11 , 27 } . We need to check if each of these numbers is an odd integer.
Checking Elements of R Let's examine each element of R :
3 is an odd integer since 3 = 2 ( 1 ) + 1 .
7 is an odd integer since 7 = 2 ( 3 ) + 1 .
11 is an odd integer since 11 = 2 ( 5 ) + 1 .
27 is an odd integer since 27 = 2 ( 13 ) + 1 .
Determining if R is a subset of A Since every element of R is an odd integer, and A is the set of all odd integers, it follows that every element of R is also an element of A . Therefore, R is a subset of A , which means R ⊂ A .
Final Answer The correct answer is "yes, because all the elements of set R are in set A".
Examples
Understanding subsets is crucial in database management. Imagine you have a database of all customers (set U). Set A represents customers who have made a purchase in the last month (active customers). Set R represents a small group of VIP customers. If all VIP customers are also active customers, then R is a subset of A. This helps in targeted marketing and personalized services.
