Problem 6 - Calculus for Biology and Medicine

Question

Assume that $$ \Omega=[1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\) Find \(A^{c}\) and show that \(\left(A^{c}\right)^{c}=A\).

Step-by-Step Solution

Answer

\(A^c = \{2, 4, 6\}\) and \((A^c)^c = A = \{1, 3, 5\}\).

Step 1: Identify Complement of Set A

The complement of set \( A \), denoted as \( A^c \), contains all the elements in the sample space \( \Omega \) that are not in \( A \). Given \( A = \{1, 3, 5\} \) and \( \Omega = \{1, 2, 3, 4, 5, 6\} \), the elements not in \( A \) are \( \{2, 4, 6\} \). Thus, \( A^c = \{2, 4, 6\} \).

Step 2: Find Complement of Complement of A

To find \( (A^c)^c \), we determine the complement of \( A^c \) within the universal set \( \Omega \). Since \( A^c = \{2, 4, 6\} \), the elements that are not in \( A^c \) are \( \{1, 3, 5\} \), which are precisely the elements of \( A \). Therefore, \( (A^c)^c = A = \{1, 3, 5\} \).

Key Concepts

Complement of a Set

The complement of a set is a crucial concept in set theory. To understand the complement of a set, let's consider a universal set or sample space, which contains all possible elements related to a problem or situation. In our example, this universal set is denoted as \( \Omega = \{1, 2, 3, 4, 5, 6\} \). When dealing with the complement of a set \( A \), we are essentially looking for all elements that are in the universal set but not in \( A \) itself.
In mathematical terms, the complement of set \( A \) is written as \( A^c \), and it includes everything in \( \Omega \) that is not present in \( A \). For instance, if \( A = \{1, 3, 5\} \), then its complement, \( A^c \), would be \( \{2, 4, 6\} \), since these are the elements of \( \Omega \) that are excluded from \( A \).

• The concept of the complement is used extensively in probability, logic, and various fields of mathematics to understand what's outside of a particular set.
• In problems involving multiple sets, computing the complement of each set can help identify intersections and unions in relation to a larger sample space.

Universal Set

The universal set is a key element of set theory that serves as the 'universe' for a particular discussion or problem space. It contains all objects under consideration and is often denoted by the symbol \( \Omega \).
For instance, in our example, the universal set \( \Omega \) contains the numbers \( \{1, 2, 3, 4, 5, 6\} \). This set is deemed complete in the context of our problem, meaning it includes every possible outcome or element we're interested in analyzing.

• Understanding what makes up your universal set is essential because all other sets, such as \( A \text{ and } B \), are subsets of this larger set.
• When dealing with complements, each set's complement must be considered in the context of the universal set.
• The universal set can vary depending on the context. For a problem involving dice, like our example, the universal set would include all possible die results, \( \{1, 2, 3, 4, 5, 6\} \).

Sample Space

In probability and statistics, a sample space, similar to a universal set, encompasses all possible outcomes of an experiment or event. It is foundational for predicting probabilities and understanding potential results.
Consider a die being rolled; the sample space is \( \{1, 2, 3, 4, 5, 6\} \), which includes all possible outcomes when you roll a standard six-sided die. In our exercise, this sample space is synonymous with the universal set \( \Omega \).

• The concept of sample space is pivotal when calculating probabilities since the likelihood of any particular event occurring is determined relative to the total number of outcomes in the sample space.
• The term "event" in probability is any subset of the sample space. For example, landing on an even number when rolling a die can be considered an event, derived from the sample space \( \Omega \).
• Identifying the sample space at the beginning of a problem-solving process provides clarity and sets boundaries for what outcomes are considered possible.