Let the universal set be the natural numbers from 40 to 60 inclusive. The universal set contains the following subsets:
- [tex]$F=$[/tex] {multiples of 6 }
- [tex]$T=$[/tex] {multiple of 2 }
True or False.
[tex]$F \subset T$[/tex]?
Answer (2)
Identify the sets: F (multiples of 6 between 40 and 60) and T (multiples of 2 between 40 and 60).
List the elements: F = { 42 , 48 , 54 , 60 } and T = { 40 , 42 , 44 , 46 , 48 , 50 , 52 , 54 , 56 , 58 , 60 } .
Check if all elements of F are in T .
Conclude that F ⊂ T is True .
Explanation
Problem Analysis Let's analyze the problem. We are given a universal set of natural numbers from 40 to 60 inclusive. We have two subsets:
F = {multiples of 6}
T = {multiples of 2}
We need to determine if F is a subset of T , which means we need to check if every element in F is also an element in T .
Listing Elements of F First, let's list the elements of set F , which are the multiples of 6 between 40 and 60 inclusive.
The multiples of 6 are: 42, 48, 54, and 60. So, F = { 42 , 48 , 54 , 60 } .
Listing Elements of T Next, let's list the elements of set T , which are the multiples of 2 between 40 and 60 inclusive.
The multiples of 2 are: 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, and 60. So, T = { 40 , 42 , 44 , 46 , 48 , 50 , 52 , 54 , 56 , 58 , 60 } .
Checking Subset Condition Now, we check if every element in F is also in T .
42 is in T .
48 is in T .
54 is in T .
60 is in T .
Since every element in F is also in T , we can conclude that F is a subset of T .
Final Answer Therefore, the statement F ⊂ T is True.
Examples
Understanding subsets helps in organizing data. For example, if you have a list of all students in a school (the universal set), you can create subsets like 'students who play basketball' or 'students who are in the math club'. If all students in the math club are also in the science club, then the math club is a subset of the science club. This kind of set theory is used in databases, computer science, and many other fields to organize and analyze information efficiently.
The statement F ⊂ T is True because all multiples of 6 between 40 and 60 are also multiples of 2 in that range.
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