To solve the problem shown in the image, I'll first transcribe and interpret the mathematical content.

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Problem:

Given the sets $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$, identify the set that represents $A \cup B$, which is the union of sets $A$ and $B$.

Choices:

A. $\{1, 2\}$
B. $\{2, 3\}$
C. $\{1, 2, 3, 4\}$
D. $\{1, 3\}$

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Solution:

The union of two sets $A$ and $B$ (denoted as $A \cup B$) is a set that contains all the elements from both sets $A$ and $B$, without repeating any element.

Given:

• $A = \{1, 2, 3\}$
• $B = \{2, 3, 4\}$

Now, let's find $A \cup B$:

$A \cup B = \{1, 2, 3\} \cup \{2, 3, 4\} = \{1, 2, 3, 4\}$

This is because:

• 1 is in $A$
• 2 is in both $A$ and $B$
• 3 is in both $A$ and $B$
• 4 is in $B$

Thus, the correct answer is:

C. $\{1, 2, 3, 4\}$

Would you like more details or have any further questions?

Here are five related questions to expand on this topic:

1. What is the difference between the union and intersection of two sets?
2. How do you find the intersection of two sets $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$?
3. Can the union of two sets ever result in a set smaller than either of the original sets?
4. How does the concept of set union extend to more than two sets?
5. What is the difference between a union of sets and a Cartesian product of sets?

Tip: The union of two sets always contains all elements from both sets, with duplicates removed.