Problem 10 - 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 \(D=\) \(\\{0,3,6,9\\}\). $$ (A \cup B) \cap(C \cup D) $$

Step-by-Step Solution

Answer

{0, 1, 6}

Step 1 - Understand Set Operations

Recall that \( \cup\) represents the union of two sets (all elements from both sets), and \( \cap\) represents the intersection of two sets (only elements common to both sets).

Step 2 - Find the Union of Sets A and B

List all unique elements in sets A and B.\[ A \cup B = {0, 2, 4, 6, 8} \cup {1, 2, 4, 8} = {0, 1, 2, 4, 6, 8} \]

Step 3 - Find the Union of Sets C and D

List all unique elements in sets C and D.\[ C \cup D = {1, 3, 5, 7, 9} \cup {0, 3, 6, 9} = {0, 1, 3, 5, 6, 7, 9} \]

Step 4 - Find the Intersection of the Results

Determine which elements are in both unions.\[ (A \cup B) \cap (C \cup D) = {0, 1, 2, 4, 6, 8} \cap {0, 1, 3, 5, 6, 7, 9} = {0, 1, 6} \]

Key Concepts

Union of Sets

The union of sets combines all the unique elements from two or more sets. The union is denoted by the symbol \( \cup \).

For example, if we take sets A and B given in the exercise:

• Set A = \( \{0,2,4,6,8\} \)
• Set B = \( \{1,2,4,8\} \)

To find the union A \( \cup \) B, we list every element that appears in either A, B, or both: \[ A \cup B = \{0, 1, 2, 4, 6, 8\} \] Notice how each element is listed only once, even if it appears in both sets.

Sets C and D can be unioned similarly:

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

Thus, their union is: \[ C \cup D = \{0, 1, 3, 5, 6, 7, 9\} \] By combining the elements once from both sets C and D.

Intersection of Sets

The intersection of sets finds the common elements between two sets. The intersection is denoted by the symbol \( \cap \).

Let's use the results of our previous union operations to determine their intersection:

• \( A \cup B = \{ 0,1,2,4,6,8 \} \)
• \( C \cup D = \{ 0,1,3,5,6,7,9 \} \)

We identify the numbers that are in both unions: numbers that appear in both results are 0, 1, and 6. Thus,
\[ (A \cup B) \cap (C \cup D) = \{0, 1, 6\} \]
So, the intersection set will contain only these common elements.

Element Identification

Identifying elements correctly within sets is crucial for performing proper set operations.

When you list out elements in a set:

• Make sure there are no duplicates.
• Check for membership within the set using clear criteria.
• Keep in mind the actual elements when combining sets, whether it's union or intersection.

For example, in our problem, we handled sets A, B, C, and D carefully to ensure no repeats and accurate inclusion:

• A = \( \{0,2,4,6,8\} \)
• B = \( \{1,2,4,8\} \)
• C = \( \{1,3,5,7,9\} \)
• D = \( \{0,3,6,9\} \)

Understanding each element's presence helps accurately perform unions and intersections.
When we combined and then intersected, careful identification ensured the correct results.