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]$F=\{$ multiples of 6$\}[/tex]
- [tex]$T=\{$ multiple of 2 $\}[/tex]
True or False.
[tex]$n(T)+n(T \prime)=21 ?$[/tex]
Answer (2)
Determine the number of elements in the universal set U : n ( U ) = 60 − 40 + 1 = 21 .
Apply the property of sets and complements: n ( T ) + n ( T ′ ) = n ( U ) .
Substitute the value of n ( U ) : n ( T ) + n ( T ′ ) = 21 .
Conclude that the statement is True .
Explanation
Problem Analysis The universal set U consists of natural numbers from 40 to 60 inclusive. The subset T contains multiples of 2 within U , and T ′ is the complement of T in U . We need to determine if the statement n ( T ) + n ( T ′ ) = 21 is true or false.
Calculating n(U) The number of elements in the universal set U , denoted as n ( U ) , is calculated as 60 − 40 + 1 = 21 .
Applying Set Properties By the properties of sets and complements, we know that n ( T ) + n ( T ′ ) = n ( U ) . Therefore, n ( T ) + n ( T ′ ) = 21 .
Conclusion Since n ( T ) + n ( T ′ ) = n ( U ) = 21 , the statement n ( T ) + n ( T ′ ) = 21 is true.
Examples
Imagine you have a group of students, and you want to divide them into two groups: those who like pizza and those who don't. If you count the number of students who like pizza and add it to the number of students who don't like pizza, you will get the total number of students in the group. This is similar to the concept of sets and complements. Let's say you have 21 students in total. If 10 students like pizza, then 21 - 10 = 11 students don't like pizza. So, 10 (students who like pizza) + 11 (students who don't like pizza) = 21 (total students). This illustrates how a set and its complement always add up to the total number of elements in the universal set.
The statement n ( T ) + n ( T ′ ) = 21 is true because there are 21 natural numbers from 40 to 60. We calculated that there are 11 multiples of 2 and 10 non-multiples of 2 within this range, which add up to 21. Therefore, the equality holds as per set properties.
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