Problem 14 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(\{1, 6, 4\}\)

Step 1: Identify the Elements of Sets C and D

Set C is given as \(C=\{1,6\}\) and set D is given as \(D=\{4\}\).

Step 2: Understand the Union Operation

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

Step 3: Combine the Elements

Combine the elements of set C and set D to form the union. List each element only once: \(\{1, 6\} \cup \{4\} = \{1, 6, 4\}\).

Step 4: Verify the Union Set

Ensure the combined set includes all elements from both sets without any repetition: The final set is \(\{1, 6, 4\}\).

Key Concepts

Set theory

Set theory is a fundamental concept in mathematics, focusing on the collection of objects known as sets. These objects can be anything – numbers, symbols, or even other sets.

Each object in a set is called an element or member of the set. For instance, in the set \(A=\{1,2,3,4,5,6\}\), the numbers 1, 2, 3, 4, 5, and 6 are elements of set A.

Sets are usually denoted by capital letters, and elements by lowercase letters or numbers. There are various operations we can perform on sets like union, intersection, and difference, each having its own rules and properties.

Union operation

The union operation is one of the basic operations in set theory. When we find the union of two sets, we combine all the elements from both sets into one single set without any duplicates.

Denoted as \(C \cup \ D\), the union of sets C and D involves merging their elements. For instance, with \(C=\{1, 6\}\) and \(D=\{4\}\), the union \(C \cup \ D\) contains all elements from both sets: \(\{1,6\} \cup \{4\} = \{1,6,4\}\). Notice that each element is listed just once.

No element repeats in the union set, even if it appears in both original sets. This rule simplifies the process of combining sets and understanding their collective elements.

Basic algebra

Basic algebra is a cornerstone of mathematics, forming the basis for more complex topics. It involves working with variables, constants, and mathematical operations like addition, subtraction, multiplication, and division.

In terms of set theory, algebra can help us better understand the properties and operations of sets. For example, the union operation is similar to addition in algebra because it combines elements, but without repetition.

With our example sets, understanding the algebraic concept behind ‘union’ can make these operations intuitive. Thinking of the union of sets \(C \cup \ D\) as a combination of their ‘elements’ helps see how fundamental algebraic rules apply to set operations. This foundational knowledge is essential for deeper mathematical studies and real-world problem solving.