Problem 12 - 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. $$ B \cup D $$

Step-by-Step Solution

Answer

The union \(B \cup D\) is \(\{1, 3, 5, 4\}\).

Step 1 - Understand the Union Operation

In set theory, the union of two sets is a set that contains all the elements from both sets, without duplication. The union is denoted by the symbol \( \cup \).

Step 2 - List Elements of Set B

Set \(B\) contains the elements \(\{1, 3, 5\}\).

Step 3 - List Elements of Set D

Set \(D\) contains the element \(\{4\}\).

Step 4 - Combine Elements of Both Sets

To find the union \(B \cup D\), combine all elements from both sets: \(\{1, 3, 5\} \cup \{4\}\).

Step 5 - Remove Duplicates and List Result

Since there are no duplicates between sets \(B\) and \(D\), the union is \(\{1, 3, 5, 4\}\).

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with studying collections of objects, known as sets. A set is simply a collection of distinct elements. These elements can be numbers, letters, or any objects. Each element in a set is unique, and the order of elements in a set does not matter. For example, the set \(A = \{1, 2, 3, 4, 5, 6\}\) contains six elements. In set theory, several operations help us analyze and manipulate sets, such as union, intersection, and difference.
When working with sets, it is important to understand these operations to solve problems that involve comparing and combining different sets.
In this exercise, we'll focus on the union operation to find the union of set \(B = \{1, 3, 5\}\) and set \(D = \{4\}\).

Union Operation

The union operation is one of the key operations in set theory. It is used when we want to combine all elements from two or more sets into one set, without any duplicates. The union of two sets A and B is denoted as \(A \cup B\).
Here's how the union operation works step-by-step:

• First, list all the elements of the first set.
• Second, list all the elements of the second set.
• Combine the elements of both sets, ensuring not to duplicate any element. Every element should only be counted once.

Let’s apply this to our example with sets B and D:

• The elements of set B are \(\{1, 3, 5\}\).
• The elements of set D are \(\{4\}\).
• Combining these elements, we get \(\{1, 3, 5\} \cup \{4\} = \{1, 3, 5, 4\}\).

The result is a new set containing all unique elements from both sets.

Element Duplication

Element duplication is the concept of ensuring that no element appears more than once in a set. In set theory, a set is defined by its unique elements, meaning no two identical elements can exist within a set.
When performing the union operation, it's crucial to remove any duplicated elements to maintain the integrity of a set.
In our step-by-step solution for the union of sets B and D, we noted that both sets contained unique elements that did not overlap. This made our task easier, as we didn't need to remove any duplicates:

• Set B elements: \(\{1, 3, 5\}\)
• Set D elements: \(\{4\}\)
• Union of B and D: \(\{1, 3, 5, 4\}\)

Hence, each element from sets B and D appears exactly once in the union.
This principle is essential for handling more complex sets where some elements may be repeated. Always check for duplicates and ensure they are removed to create a proper union set.