Problem 24 - 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. $$ C \cup D $$

Step-by-Step Solution

Answer

The union of sets C and D is \(-3, -1, 0, 1, 2, 5, 8\).

Step 1: Understand the Union

The union of two sets (denoted as \(C \cup D\)) is a set that contains all elements from both sets without any duplicates. To find the union of sets \(C\) and \(D\), list all unique elements from both sets.

Step 2: List Elements of Set C

Set \(C\) contains the elements: \(-3, -1, 0, 1, 2\). Keep these elements in mind as we compare with Set \(D\).

Step 3: List Elements of Set D

Set \(D\) contains the elements: \(-3, 1, 2, 5, 8\). We'll combine these with the elements from Set \(C\).

Step 4: Combine the Sets and Remove Duplicates

Take elements from both sets \(C\) and \(D\) and list them together: \(-3, -1, 0, 1, 2, 5, 8\). Notice how some elements appear in both sets, so we only list them once in the union.

Key Concepts

Union of Sets

In set theory, the concept of the union of sets is fundamental. The union of two sets combines all the elements present in either set. This is denoted using the symbol \(\cup\). The union not only brings the elements of both sets together but also ensures that each element appears only once. To illustrate, when we have two sets, \(C=\{-3,-1,0,1,2\}\) and \(D=\{-3,1,2,5,8\}\), finding the union involves:

• Listing all unique elements found in either set.
• Ensuring no duplicates in the final list.

For these specific sets, the union \(C \cup D\) results in \(-3, -1, 0, 1, 2, 5, 8\). Notice that elements \(-3\), \(1\), and \(2\) appear in both sets, but are included only once in the union.

Elementary Algebra

Elementary algebra serves as the foundation for understanding how to solve algebraic expressions and equations. Most of the operations from elementary algebra are also applicable to set theory. Essentially, operations such as union, intersection, and difference of sets have similarities to basic algebraic manipulations.
Just like in algebraic operations where you combine like terms, the union of sets involves combining elements and removing duplicates. This is analogous to simplifying algebraic expressions to get a single expression without repetition.

• In algebra, terms like \(x + x = 2x\), simplify repetition.
• In set theory, elements like \(a\) and \(a\) once combined, stay as \(a\).

Understanding these parallels can simplify learning both domains of mathematics.

Mathematical Notation

Mathematical notation is a universal language that conveys complex mathematical ideas succinctly and precisely. Notations are symbols and phrases designed to efficiently express concepts and relationships. For example, the union of sets is denoted by \( \cup \) and provides a shorthand method to describe the concept of bringing two sets together.
Each mathematical symbol carries significant meaning and helps prevent misunderstandings when communicating mathematical ideas. For sets, key notations include:

• \(A \cup B\) for the union of sets \(A\) and \(B\).
• \(A \cap B\) for the intersection, showing common elements.
• \(A - B\) for the difference, indicating elements in \(A\) not in \(B\).

Using proper notation not only makes writing mathematics clear but also aids in learning and understanding the intricate ideas behind those symbols.