### Union of Sets

#### Definition of Union
The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is the set consisting of all elements that are in set [tex]\( A \)[/tex] or set [tex]\( B \)[/tex]. This means the combination of all unique elements from set [tex]\( A \)[/tex] and set [tex]\( B \)[/tex].

#### Notation
[tex]\[ A \cup B \][/tex]
[tex]\[ A \cup B = \{ x \mid x \in A \text{ or } x \in B \} \][/tex]

#### Example
Let [tex]\( A = \{1, 2, 4\} \)[/tex], [tex]\( B = \{2, 4, 6, 8\} \)[/tex], and [tex]\( C = \{5, 7, 9\} \)[/tex]. Find the following:

a. [tex]\( A \cup B = \)[/tex]

b. [tex]\( A \cup C = \)[/tex]

c. [tex]\( B \cup C = \)[/tex]

Let's go through each part of the question step-by-step using the concept of the union of sets.

### Part (a) [tex]\( A \cup B \)[/tex]

Given:
- [tex]\( A = \{1, 2, 4\} \)[/tex]
- [tex]\( B = \{2, 4, 6, 8\} \)[/tex]

Solution:

1. List all unique elements in set [tex]\( A \)[/tex]: \{1, 2, 4\}
2. List all unique elements in set [tex]\( B \)[/tex]: \{2, 4, 6, 8\}
3. Combine the elements from both sets, ensuring no duplicates:
- Unique elements: \{1, 2, 4, 6, 8\}

Thus, [tex]\( A \cup B = \{1, 2, 4, 6, 8\} \)[/tex].

### Part (b) [tex]\( A \cup C \)[/tex]

Given:
- [tex]\( A = \{1, 2, 4\} \)[/tex]
- [tex]\( C = \{5, 7, 9\} \)[/tex]

Solution:

1. List all unique elements in set [tex]\( A \)[/tex]: \{1, 2, 4\}
2. List all unique elements in set [tex]\( C \)[/tex]: \{5, 7, 9\}
3. Combine the elements from both sets, ensuring no duplicates:
- Unique elements: \{1, 2, 4, 5, 7, 9\}

Thus, [tex]\( A \cup C = \{1, 2, 4, 5, 7, 9\} \)[/tex].

### Part (c) [tex]\( B \cup C \)[/tex]

Given:
- [tex]\( B = \{2, 4, 6, 8\} \)[/tex]
- [tex]\( C = \{5, 7, 9\} \)[/tex]

Solution:

1. List all unique elements in set [tex]\( B \)[/tex]: \{2, 4, 6, 8\}
2. List all unique elements in set [tex]\( C \)[/tex]: \{5, 7, 9\}
3. Combine the elements from both sets, ensuring no duplicates:
- Unique elements: \{2, 4, 5, 6, 7, 8, 9\}

Thus, [tex]\( B \cup C = \{2, 4, 5, 6, 7, 8, 9\} \)[/tex].

### Summary

Summarizing the results:

a. [tex]\( A \cup B = \{1, 2, 4, 6, 8\} \)[/tex]

b. [tex]\( A \cup C = \{1, 2, 4, 5, 7, 9\} \)[/tex]

c. [tex]\( B \cup C = \{2, 4, 5, 6, 7, 8, 9\} \)[/tex]

These are the unions of the given sets [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].