Problem 23 - Elementary and Intermediate Algebra

Question

Let \(A=\\{0,1,2,3,4,5,6\\}, B=\\{4,6,8,10\\}\) \(C=\\{-3,-1,0,1,2\\},\) and \(D=\\{-3,1,2,5,8\\}\) Find each set. $$ A \cup D $$

Step-by-Step Solution

Answer

The set is \(\{-3, 0, 1, 2, 3, 4, 5, 6, 8\}\).

Step 1: Understand the Union Operation

The union of two sets, denoted as \(A \cup D\), is the collection of all distinct elements from both sets. Any element that appears in either set will appear in their union.

Step 2: Identify Elements of Set A

Set \(A\) is given as \(\{0, 1, 2, 3, 4, 5, 6\}\). These are the distinct elements you will include in the union operation.

Step 3: Identify Elements of Set D

Set \(D\) is given as \(\{-3, 1, 2, 5, 8\}\). Any of these elements that are unique to set \(D\) need to be included in the union result.

Step 4: Combine the Elements

List all elements from set \(A\) and include all distinct elements from set \(D\). Combine them to form the union: \(\{-3, 0, 1, 2, 3, 4, 5, 6, 8\}\).

Step 5: Remove Duplicates

Ensure each element in the union set appears only once. The elements in \(A \cup D\) should be distinct: \(\{-3, 0, 1, 2, 3, 4, 5, 6, 8\}\).

Key Concepts

Union of Sets

One of the fundamental operations in set theory is finding the **union** of sets. When we perform a union operation, we are essentially merging two or more sets. The union of sets includes all the elements that belong to any of the sets involved.
Consider two sets, say, set \(A\) and set \(D\). The union is represented using the symbol \(\cup\). So, \(A \cup D\) signifies the union of sets \(A\) and \(D\).

• This operation takes every element from both sets.
• If an element appears in both sets, it is included only once in the union.

For our exercise, set \(A = \{0, 1, 2, 3, 4, 5, 6\}\) and set \(D = \{-3, 1, 2, 5, 8\}\), the union \(A \cup D\) results in \(\{-3, 0, 1, 2, 3, 4, 5, 6, 8\}\). Notice how elements from both sets form the complete collection in the union.

Mathematical Sets

In set theory, a **set** is a collection of distinct, well-defined objects. These objects can be anything: numbers, letters, fruits, etc.

• Each object within a set is referred to as an element.
• Sets are denoted using curly braces: \(\{\}\).

In mathematics, sets are fundamental and are used to define various concepts. For example, sets can represent collections of numbers, such as whole numbers, integers, or even more complex structures.
Sets can be compared to containers that hold their elements. For instance, in our exercise:

• Set \(A = \{0, 1, 2, 3, 4, 5, 6\}\)
• Set \(D = \{-3, 1, 2, 5, 8\}\)

These sets clearly define a collection of distinct numbers, each enclosed within its respective set brackets.

Distinct Elements

The notion of **distinct elements** is crucial when working with sets. In a set, each element must be unique, meaning no duplicates are allowed.

• Even if an element appears multiple times in the process of unions or other operations, it is listed only once in the final set.
• This property maintains the uniqueness and clarity within a set.

In the exercise, when forming the union \(A \cup D\), we combine the elements from set \(A\) and set \(D\). Elements may appear in both sets, such as \(1, 2,\) and \(5\). These are included only once in the union result to ensure all elements are distinct.
Remember, distinct means no repetitions, making it easy to assess the makeup of any given set or union of sets.