Problem 11 - 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. $$ A \cup B $$

Step-by-Step Solution

Answer

{1, 2, 3, 4, 5, 6}

Step 1 - Identify the elements of set A

Set A contains the elements: A = {1, 2, 3, 4, 5, 6}.

Step 2 - Identify the elements of set B

Set B contains the elements: B = {1, 3, 5}.

Step 3 - Understand the union operation

The union of two sets, denoted as \(A \cup B\), is the set of all distinct elements that are in either set A, set B, or both.

Step 4 - Combine the elements

Combine all the elements from set A and set B, listing each element only once: {1, 2, 3, 4, 5, 6}.

Key Concepts

sets in algebra

A set in algebra is a collection of distinct objects, considered as an entity in its own right. Sets are typically used to group numbers, variables, or other items.

Each element in a set is unique. For instance, if we have a set of numbers, no number will appear more than once.

Sets can be denoted using curly braces, like in this exercise:
\(A=\{1,2,3,4,5,6\}, B=\{1,3,5\}, C=\{1,6\}, D=\{4\}\).
Here, each variable represents a set of distinct elements. Elements in a set can have anything in common, but each element is listed only once.

• Set Notation: Sets are enclosed in curly braces \( \{ \} \).
• Element: An object in a set, e.g., 1 is an element of set A.

Understanding sets in algebra is fundamental, as they form the basis for more complex operations.

union operation

In set theory, the union operation is one way to combine sets. The union of two or more sets results in a set containing all distinct elements from each of the original sets.

The symbol for the union operation is \( \cup \).
For example, \( A \cup B \) represents the union of sets A and B.

Performing the union operation involves listing all elements from both sets and eliminating any duplicates, ensuring that each element is only included once.

• Notation: Union is denoted by \( \cup \).
• Distinct Elements: No duplicates in the resulting set.

Applying this to our exercise, we combine sets A and B as follows:
Step 1: A = {1, 2, 3, 4, 5, 6}
Step 2: B = {1, 3, 5}
Step 3: A \cup B = {1, 2, 3, 4, 5, 6}.
Even though sets A and B have common elements, the union lists each element once, ensuring no repetitions.

distinct elements

When working with sets, the term 'distinct elements' refers to elements that are unique and appear only once in the set.

For instance, in the set \( \{1,2,3,4,5,6\} \), each number from 1 to 6 is distinct.
When combining sets using union, only distinct elements should be included in the resulting set.

• Unique Appearance: Each element appears only once, even if it is repeated in the original sets.
• Combination of Sets: When combining sets \( \{1,2,3\} \) and \( \{2,3,4\} \), the result is \( \{1,2,3,4\} \), with each element unique.

This ensures clarity and precision in set operations.

In our exercise, when we perform the union of A and B,
we consider each element from both sets but write them only once to make sure we have a set of distinct elements:
\(A \cup B = \{1,2,3,4,5,6\}\).
There are no duplicate entries in the union.