Problem 6 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\) $$ A \cup C $$

Step-by-Step Solution

Answer

\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}

Step 1 - Understand Union of Sets

The union of two sets, denoted as \(A \cup C\), includes all elements that are in either set A or set C, or in both.

Step 2 - List Elements of Set A

Examine set A. The elements of set A are \(\{0, 2, 4, 6, 8\}\).

Step 3 - List Elements of Set C

Examine set C. The elements of set C are \(\{1, 3, 5, 7, 9\}\).

Step 4 - Combine the Elements

Combine all the unique elements from sets A and C. This includes \(0, 1, 2, 3, 4, 5, 6, 7, 8, \text{ and } 9\).

Step 5 - List the Union Set

List the elements of the union set \(A \cup C\). The union set is \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\.

Key Concepts

set theory

Set theory is a branch of mathematical logic that studies collections of objects, which are called sets. Sets are fundamental objects in mathematics. They can include anything: numbers, letters, or even other sets.

A set is usually denoted by a capital letter and its elements are listed within curly brackets, for example, \(A = \{1, 2, 3\}\). The elements of a set are distinct and order does not matter.

Some key operations in set theory include:

• Union
• Intersection
• Difference
• Complement

The union of two or more sets is one of the fundamental operations. It combines all elements from the involved sets.

union operation

The union operation is a way to combine two sets by including any element that is in either set or in both. The union of sets A and B is denoted as \(A \cup B \).

For instance, if we have two sets, \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), their union \(A \cup B\) would be \{1, 2, 3, 4, 5\}\. Notice that the element 3, which is common to both sets, is not repeated in the result.

Here's a step-by-step breakdown of how to find the union of sets:

• List all elements of the first set.
• List all elements of the second set.
• Combine the elements, making sure not to repeat any element.

By following these steps, you can determine the union of any two sets.

mathematical notation

Mathematical notation is a system of symbols used to express mathematical concepts and operations. It is essential for writing clear and concise mathematical expressions.

In the context of set theory and union operations, we use specific symbols to denote different operations and elements. Here are some commonly used notations:

• Curly brackets \(\{ \}\) are used to list the elements of a set. For example, \(A = \{0, 2, 4\}\).
• The union symbol \(\cup\) represents the union of two sets.
• Equals sign \(=\) states the equality of two expressions.

A good grasp of mathematical notation is crucial for understanding and performing set operations efficiently.

In our exercise, the instructions employ these notations to guide you through finding the union of two sets, \(A \cup C\).