Problem 8 - Calculus for Biology and Medicine

Question

In Problems \(5-8\), assume that $$ \Omega=\\{1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\). Are \(A\) and \(B\) disjoint?

Step-by-Step Solution

Answer

Sets \(A\) and \(B\) are not disjoint because they share elements.

Step 1: Understand the Definition of Disjoint Sets

Two sets are considered disjoint if they have no elements in common. We need to determine if there are any common elements between sets \(A\) and \(B\).

Step 2: Identify Common Elements

List the elements of both sets. Set \(A\) contains \(\{1, 3, 5\}\) and set \(B\) contains \(\{1, 2, 3\}\). Identify any elements that are present in both sets.

Step 3: Compare Elements of A and B

Observe that the elements \(1\) and \(3\) are present in both \(A\) and \(B\). Thus, these are the common elements between the two sets.

Step 4: Make Conclusion about Disjoint Status

Since \(A\) and \(B\) have common elements \(1\) and \(3\), they are not disjoint. Disjoint sets have no elements in common, and here, we have found common elements.

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that studies collections of objects, known as sets. A set is simply a collection of distinct elements, which could be anything, such as numbers, objects, or even other sets. In our context, consider the universal set \( \Omega = \{1, 2, 3, 4, 5, 6\} \), from which subsets like \( A \) and \( B \) are derived. The elements in a set are listed inside curly braces and separated by commas. Understanding this framework is crucial, as it forms the basis for many discussions in mathematics and logic. When working with sets, one typically examines the relationships between them, such as whether they overlap or are entirely separate. This leads to concepts like union, intersection, and complement, which describe how sets relate to each other.

Common Elements

Common elements between sets are those that appear in more than one set. When two sets \( A = \{1, 3, 5\} \) and \( B = \{1, 2, 3\} \) are considered, you focus on identifying elements that exist in both. In this case, we list the elements of each set:

• Set \( A \): \{1, 3, 5\}
• Set \( B \): \{1, 2, 3\}

The common elements are those that appear in both lists, which are \(1\) and \(3\). Identifying common elements helps in determining whether the sets share any overlap. If there are no common elements, the sets are disjoint; otherwise, they aren't. Recognizing common elements is often the first step in evaluating relationships between sets.

Problem Solving Steps

When solving problems related to whether two sets are disjoint, following a methodical approach can help clarify the process:

• Step 1: Understand that disjoint sets have no elements in common.
• Step 2: List the elements of each set to identify any common ones.
• Step 3: Compare these lists to find shared elements.
• Step 4: Draw a conclusion. If no shared elements are found, the sets are disjoint.

In the exercise, we evaluated sets \( A \) and \( B \) to determine disjoint status. By comparing the elements, we identified common elements \(1\) and \(3\). Hence, these sets are not disjoint, as they share elements. Using these structured steps simplifies the problem-solving process, ensuring a clear path from identifying the data to arriving at a conclusion.