Let's approach the problem step-by-step, focusing on determining the complements of the given sets within the universal set [tex]\( U \)[/tex].
- Universal Set:
[tex]\[ U = \{29, 30, 31, \ldots, 45\} \][/tex]
a) Complement of [tex]\(A\)[/tex]:
Given:
[tex]\[ A = \{32, 33, 37, 39, 42, 44\} \][/tex]
The complement of [tex]\( A \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
So, let's list the elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]:
[tex]\[
A^c = U - A = \{29, 30, 31, 34, 35, 36, 38, 40, 41, 43, 45\}
\][/tex]
Therefore,
[tex]\[ \boxed{\{34, 35, 36, 38, 40, 41, 43, 45, 29, 30, 31\}} \][/tex]
b) Complement of [tex]\(B\)[/tex]:
Given:
[tex]\[ B = \{ \text{multiples of 4 within } U \} \][/tex]
The multiples of 4 in the set [tex]\( U \)[/tex] are:
[tex]\[ B = \{32, 36, 40, 44\} \][/tex]
The complement of [tex]\( B \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( B \)[/tex].
So, let's list the elements in [tex]\( U \)[/tex] that are not in [tex]\( B \)[/tex]:
[tex]\[
B^c = U - B = \{29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45\}
\][/tex]
Therefore,
[tex]\[ \boxed{\{33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 29, 30, 31\}} \][/tex]
c) Complement of [tex]\(C\)[/tex]:
Given:
[tex]\[ C = \{ \text{non-prime numbers within } U \} \][/tex]
We first need to identify the non-prime numbers within the set [tex]\( U \)[/tex]:
Non-prime numbers in [tex]\( U \)[/tex] are [tex]\( \{30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45\} \)[/tex].
Thus,
[tex]\[ C = \{30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45\} \][/tex]
The complement of [tex]\( C \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( C \)[/tex].
So, let's list the elements in [tex]\( U \)[/tex] that are not in [tex]\( C \)[/tex]:
[tex]\[
C^c = U - C = \{29, 31, 37, 41, 43\}
\][/tex]
Therefore,
[tex]\[ \boxed{\{37, 41, 43, 29, 31\}} \][/tex]
In summary, the complements of the given sets are:
a) [tex]\( A^c = \boxed{\{34, 35, 36, 38, 40, 41, 43, 45, 29, 30, 31\}} \)[/tex]
b) [tex]\( B^c = \boxed{\{33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 29, 30, 31\}} \)[/tex]
c) [tex]\( C^c = \boxed{\{37, 41, 43, 29, 31\}} \)[/tex]
