Problem 5 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\(A \cup B = \{1, 2, 3, 5\}\) and \(A \cap B = \{1, 3\}\).

Step 1: Understanding Notations

In set theory, given sets, the notation \( A \cup B \) represents the union of sets \( A \) and \( B \). This refers to a set containing all elements that belong to either set \( A \), or set \( B \), or both. The notation \( A \cap B \) represents the intersection of sets \( A \) and \( B \). This refers to a set of all elements that are common to both sets \( A \) and \( B \).

Step 2: Finding the Union of Sets A and B

The union of the sets \(A\) and \(B\), denoted \(A \cup B\), consists of all elements that are in either \(A\) or \(B\) or in both. Starting with set \(A = \{1, 3, 5\}\) and \(B = \{1, 2, 3\}\), we compile all unique elements. Therefore, \(A \cup B = \{1, 2, 3, 5\}\).

Step 3: Finding the Intersection of Sets A and B

The intersection of the sets \(A\) and \(B\), denoted \(A \cap B\), consists of all elements that are common to both sets. Looking at \(A = \{1, 3, 5\}\) and \(B = \{1, 2, 3\}\), the common elements are 1 and 3. Thus, \(A \cap B = \{1, 3\}\).

Key Concepts

Union of Sets

In set theory, when we talk about the union of two sets, we are essentially combining all the elements from both sets into one big set. The resulting set consists of all elements that appear in either of the sets or in both. This operation is represented by the symbol \( \cup \). For example, consider sets \( A = \{1, 3, 5\} \) and \( B = \{1, 2, 3\} \). The union \( A \cup B \) includes every element present in either set. We gather all these elements as follows:

• From set \( A \): 1, 3, 5
• From set \( B \): 1, 2, 3

When we list them without repeating any element, our union is \( \{1, 2, 3, 5\} \). Notice how we included number 2 from set \( B \) and number 5 from set \( A \), even though they are present in only one of the sets. Each element appears only once in our final union set.

Intersection of Sets

The intersection of sets is another fundamental concept in set theory. It is denoted by the symbol \( \cap \) and involves finding a common ground between two sets. Specifically, the intersection set includes only those elements that are present in both sets. Imagine sets \( A = \{1, 3, 5\} \) and \( B = \{1, 2, 3\} \). Here, our task is to identify which numbers appear in both sets at the same time.

• Elements in both \( A \) and \( B \): 1, 3

Thus, the intersection \( A \cap B \) results in the set \( \{1, 3\} \), showcasing how details of each set overlap with one another by focusing only on shared elements. The concept of intersection effectively narrows down the combined scope of the two sets to pinpoint exactly which elements they share.

Element Notation

Element notation is a simple but key aspect of understanding sets and dealing with operations like union and intersection. When working with sets, it's crucial to know how elements within these sets are represented and identified. Elements in a set are typically listed within curly brackets \( \{ \} \), and they are separated by commas to distinguish each element clearly. For instance, if we encounter a set \( A = \{1, 3, 5\} \), we easily spot that 1, 3, and 5 are discrete and separate elements of set \( A \). When discussing membership, we use specific symbols:

• \( x \in A \): Element \( x \) is in set \( A \)
• \( y otin A \): Element \( y \) is not in set \( A \)

Through this notation, not only can we indicate which elements belong to a set, but we can also carry out operations involving these sets, such as finding unions and intersections with greater ease and accuracy.