Problem 5 - 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 \(A \cup B\) and \(A \cap B\).

Step-by-Step Solution

Answer

The union is \( \{1, 2, 3, 5\} \) and the intersection is \( \{1, 3\} \).

Step 1: Understand the Union

The union of two sets, denoted as \(A \cup B\), consists of all elements from both sets, without duplicates. Essentially, it includes every element that is in set \(A\), set \(B\), or both.

Step 2: Calculate the Union

List the elements from sets \(A\) and \(B\):- Set \(A\) has elements \(\{1, 3, 5\}\).- Set \(B\) has elements \(\{1, 2, 3\}\).The union, \(A \cup B\), includes each unique element from both sets:- Union Result: \(\{1, 2, 3, 5\}\).

Step 3: Understand the Intersection

The intersection of two sets, denoted as \(A \cap B\), consists of only the elements that are present in both sets \(A\) and \(B\).

Step 4: Calculate the Intersection

Identify common elements in sets \(A\) and \(B\):- Compare both sets: - \(1\) is in both \(A\) and \(B\). - \(3\) is in both \(A\) and \(B\).The intersection, \(A \cap B\), consists of these shared elements:- Intersection Result: \(\{1, 3\}\).

Key Concepts

Union of Sets

A union of two sets brings together all the elements from each set into one collective set. When we talk about the union, denoted as \(A \cup B\), it includes all elements from set \(A\), all elements from set \(B\), while ensuring no repetition of elements.
This means if an element appears in both sets, it is still listed only once in the union.
For example, considering our sets:

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

To find \(A \cup B\), you combine all distinct elements from both sets:

• \(1\): present in both \(A\) and \(B\)
• \(2\): present in \(B\) only
• \(3\): present in both \(A\) and \(B\)
• \(5\): present in \(A\) only

Therefore, the union \(A \cup B\) results in \(\{1, 2, 3, 5\}\). It's all about merging the elements from each set into one whole, showing every piece of data without overlap.

Intersection of Sets

The intersection of two sets focuses on finding elements that two sets have in common. When you hear about the intersection, represented as \(A \cap B\), it means you are looking only for elements found in both sets simultaneously.
Using our example sets:

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

We want to determine which numbers occur in both \(A\) and \(B\).
To find \(A \cap B\), you'll look for common values:

• \(1\): present in both \(A\) and \(B\)
• \(3\): also present in both \(A\) and \(B\)

The intersection \(A \cap B\) results in \(\{1, 3\}\). This step narrows down to include only shared elements, a common ground where both sets overlap.

Element of a Set

An element of a set refers to a singular item or member that is part of a set. Understanding this concept is crucial as it forms the basis of working with sets in set theory. Each element in a set is unique, and a set can include numbers, symbols, or even other sets.
Key points about elements include:

• Elements within a set are written in curly brackets, like \(\{1, 2, 3\}\).
• Each element appears uniquely; there are no duplicates inside a set.
• A set does not consider the order, thus \(\{1, 2, 3\}\) is equivalent to \(\{3, 2, 1\}\).

The concept is quite straightforward: if something is part of a defined collection, it's considered an element of that set. For instance, in the set \(\{1, 3, 5\}\), the number \(1\) is an element. Being part of the set means it meets the criteria that define what's inside the grouping.