Sure! Let's solve this step-by-step.
1. Define the Universal Set:
The universal set [tex]\( U \)[/tex] is the set of natural numbers. For the sake of simplicity, we can consider the first 100 natural numbers (i.e., [tex]\( \{1, 2, 3, \ldots, 100\} \)[/tex]).
2. Define Set [tex]\( A \)[/tex]:
Set [tex]\( A \)[/tex] is defined as the set of all odd natural numbers within the universal set. So, [tex]\( A \)[/tex] will contain all the odd numbers from 1 to 100. Specifically:
[tex]\[
A = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99\}
\][/tex]
3. Complement of Set [tex]\( A \)[/tex] (denoted [tex]\( A^{\prime} \)[/tex]):
The complement of set [tex]\( A \)[/tex] within the universal set [tex]\( U \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]. In other words, [tex]\( A^{\prime} \)[/tex] will be the set of all even natural numbers from 1 to 100. Therefore, [tex]\( A^{\prime} \)[/tex] will contain:
[tex]\[
A^{\prime} = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100\}
\][/tex]
To summarize:
- The set [tex]\( A \)[/tex] (odd natural numbers within 1 to 100) is:
[tex]\[
A = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99\}
\][/tex]
- The complement set [tex]\( A^{\prime} \)[/tex] (even natural numbers within 1 to 100) is:
[tex]\[
A^{\prime} = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100\}
\][/tex]
