Sure, let's work through each part of the question step by step to find the complements of the given sets.
Part (a): Complement of [tex]\( A \)[/tex]
Given:
- The universal set [tex]\( U = \{29, 30, 31, \ldots, 45\} \)[/tex].
- The set [tex]\( A = \{32, 33, 37, 39, 42, 44\} \)[/tex].
To find the complement of [tex]\( A \)[/tex] (denoted as [tex]\( A^c \)[/tex]), we need to include all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
The elements in [tex]\( U \)[/tex] are: [tex]\( 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45 \)[/tex].
Removing the elements of [tex]\( A \)[/tex] from [tex]\( U \)[/tex]:
[tex]\[
A^c = U - A = \{29, 30, 31, 34, 35, 36, 38, 40, 41, 43, 45\}
\][/tex]
Part (b): Complement of [tex]\( B \)[/tex]
Given:
- The set [tex]\( B = \{x \in U : x \text{ is a multiple of 4} \}\)[/tex].
Multiples of 4 within the range of [tex]\( U \)[/tex] are: [tex]\( 32, 36, 40, 44 \)[/tex].
To find the complement of [tex]\( B \)[/tex] (denoted as [tex]\( B^c \)[/tex]), we include all elements in [tex]\( U \)[/tex] that are not multiples of 4.
[tex]\[
B^c = U - B = \{29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45\}
\][/tex]
Part (c): Complement of [tex]\( C \)[/tex]
Given:
- The set [tex]\( C \)[/tex] includes non-prime numbers in [tex]\( U \)[/tex].
Non-prime numbers within the range of [tex]\( U \)[/tex] are: [tex]\( 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45 \)[/tex].
To find the complement of [tex]\( C \)[/tex] (denoted as [tex]\( C^c \)[/tex]), we include all elements in [tex]\( U \)[/tex] that are prime numbers.
The prime numbers in the range [tex]\( U \)[/tex] are: [tex]\( 29, 31, 37, 41, 43 \)[/tex].
[tex]\[
C^c = U - C = \{29, 31, 37, 41, 43\}
\][/tex]
Summary of Complements:
- [tex]\( A^c = \{34, 35, 36, 38, 40, 41, 43, 45, 29, 30, 31\}\)[/tex]
- [tex]\( B^c = \{33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 29, 30, 31\} \)[/tex]
- [tex]\( C^c = \{37, 41, 43, 29, 31\} \)[/tex]
Thus, those are the complements for each of the sets [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
