Problem 2 - 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 \(D=\) \(\\{0,3,6,9\\}\) $$ C \cup D $$

Step-by-Step Solution

Answer

\( C \cup D = \{0,1,3,5,6,7,9\} \)

Step 1 - Understand Union of Sets

To find the union of two sets, combine all the elements from both sets without repeating any element. The symbol for union is \( \cup \) which represents 'or'.

Step 2 - Identify Elements of Both Sets

List the elements of sets C and D.\( C = \{1,3,5,7,9\}\) and \( D = \{0,3,6,9\}\)

Step 3 - Combine Elements

Combine the elements from both sets without repeating any element: \( \{1,3,5,7,9\} \cup \{0,3,6,9\} = \{0,1,3,5,6,7,9\} \)

Key Concepts

set theory

Set theory is a fundamental branch of mathematics that deals with collections of objects, which we call 'sets'. A set can contain numbers, letters, or even other sets. The study of set theory helps in understanding various mathematical concepts like functions, relations, and more complex structures.
Sets are usually represented by curly braces containing their elements. For example, the set of even numbers less than 10 can be written as \( \{0, 2, 4, 6, 8\} \).
Understanding set theory is crucial for grasping other mathematical concepts, as it forms the foundation for more advanced topics.

elements of a set

Elements of a set are the individual objects or members contained within that set. Each element in a set is unique; that means no element is repeated.
For example, if we have a set \( A = \{0, 2, 4, 6, 8\} \), then 0, 2, 4, 6, and 8 are the elements of set A.
Understanding the elements of a set is important for performing operations like union, intersection, and difference. It helps in identifying what belongs to a set and what doesn't. This is the basic step before performing any set operation.

union symbol

The union symbol \( \cup \) is used in set theory to denote the union of two sets. The union of sets A and B consists of all the elements that are in set A, in set B, or in both.
For example, if we have sets \( C = \{1, 3, 5, 7, 9\} \) and \( D = \{0, 3, 6, 9\} \), the union of sets C and D is written as \( C \cup D \).
To find the union: Combine all unique elements from both sets: \( C \cup D = \{0, 1, 3, 5, 6, 7, 9\} \). Notice that elements like 3 and 9, which are present in both sets, appear only once in the union.