Use the following to answer the next 4 questions.
Let the universal set be the natural numbers from 40 to 60 inclusive. The universal set contains the following subsets:
- [tex]F=\{\text{multiples of 6}\}[/tex]
- [tex]T=\{\text{multiple of 2}\}[/tex]
True or False.
[tex]T I=\{m \mid m=2 x, 20 \leq x \leq 30, x \in R\} ??[/tex]
Answer (2)
Define the universal set as natural numbers from 40 to 60.
Identify set T as multiples of 2 within the universal set: T = { 40 , 42 , 44 , 46 , 48 , 50 , 52 , 54 , 56 , 58 , 60 } .
Define interval I as { m ∣ m = 2 x , 20 ≤ x ≤ 30 , x ∈ R } , which is the interval [ 40 , 60 ] .
Determine that T ∩ I is the set of even integers from 40 to 60, which is not equal to the interval I .
Conclude that the statement is F a l se .
Explanation
Understanding the Problem The universal set consists of natural numbers from 40 to 60, inclusive. We have two subsets: F , which contains multiples of 6, and T , which contains multiples of 2. The question asks whether the intersection of T and the interval I = { m ∣ m = 2 x , 20 ≤ x ≤ 30 , x ∈ R } is equal to the interval I itself.
Identifying Set T First, let's identify the elements of set T . Since T contains multiples of 2 within the universal set (40 to 60), T = { 40 , 42 , 44 , 46 , 48 , 50 , 52 , 54 , 56 , 58 , 60 } .
Analyzing Interval I Next, let's analyze the interval I = { m ∣ m = 2 x , 20 ≤ x ≤ 30 , x ∈ R } . This interval represents all real numbers m that can be expressed as 2 x , where x ranges from 20 to 30, inclusive. Therefore, I is the interval [ 40 , 60 ] .
Finding the Intersection of T and I Now, we need to find the intersection of T and I , denoted as T ∩ I . Since T contains only even integers from 40 to 60, and I contains all real numbers from 40 to 60, their intersection will be the set of even integers from 40 to 60. Thus, T ∩ I = { 40 , 42 , 44 , 46 , 48 , 50 , 52 , 54 , 56 , 58 , 60 } .
Comparing the Sets The question asks if T ∩ I = I . However, T ∩ I is the set of even integers from 40 to 60, while I is the interval of all real numbers from 40 to 60. These two sets are not equal because I includes all real numbers between 40 and 60 (e.g., 40.5, 41.7, etc.), which are not in T ∩ I .
Final Answer Therefore, the statement T ∩ I = { m ∣ m = 2 x , 20 ≤ x ≤ 30 , x ∈ R } is false.
Examples
Understanding sets and intervals is crucial in many real-world scenarios. For example, in data analysis, you might have a dataset representing customer ages. The universal set could be all possible ages, while subsets could represent specific age groups (e.g., teenagers, adults, seniors). Intervals could represent age ranges for targeted marketing campaigns. Determining the intersection of these sets and intervals helps you identify the exact group of customers that meet specific criteria, allowing for more effective marketing strategies.
The statement that the intersection of set T and the interval I is equal to the interval I is false. Set T contains only the even numbers between 40 and 60, while interval I includes all real numbers in that range. Therefore, T ∩ I does not equal I.
;
