Finding The Number Of Elements In Set Union A ∪ B

by ADMIN 50 views

Introduction

Hey guys! Today, we're diving into a fun problem from set theory. We've got two sets, A and B, and our mission is to find the number of elements in their union, denoted as n(A ∪ B). This might sound a bit complex, but don't worry, we'll break it down step by step. Set theory is a fundamental concept in mathematics, and mastering it can help you tackle a wide range of problems, from basic counting to more advanced topics in discrete mathematics and computer science. Understanding how sets interact, especially the union of sets, is crucial for various applications, including database management, data analysis, and algorithm design. So, let's get started and unravel this interesting problem together!

Understanding the Basics of Sets

Before we jump into the problem, let's quickly recap what sets and unions are all about. A set is simply a collection of distinct objects, which we call elements or members. For example, set A contains prime numbers, and set B contains integers from 1 to 10. The union of two sets, denoted by A ∪ B, is a new set that contains all the elements that are in A, in B, or in both. It's like merging the two sets into one big group, but we only count each element once, even if it appears in both sets. Think of it as inviting all the members of two clubs to a party, but you only want to count each person once to get an accurate headcount.

The Problem at Hand

We're given two sets:

  • A = {2, 3, 5, 7, 11, 13, 17}
  • B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Our goal is to find n(A ∪ B), which means we need to count the number of unique elements in the union of sets A and B. To solve this, we'll first find the union of the two sets, and then simply count how many elements are in that combined set. This might seem straightforward, but it's important to be meticulous and avoid counting any element twice. We'll explore a couple of different ways to approach this, ensuring we arrive at the correct answer. Understanding the problem thoroughly is the first step towards finding the solution, and in this case, it's all about merging the two sets and counting the unique elements.

Solution

Step-by-Step Solution

Alright, let's get down to business and solve this problem step by step. First, we need to find the union of sets A and B, which means we'll combine all the elements from both sets into a single set, making sure not to repeat any elements. This is like merging two lists, but if an item appears on both lists, we only include it once in our final merged list.

  1. List the Elements of Set A: Set A contains the elements {2, 3, 5, 7, 11, 13, 17}.
  2. List the Elements of Set B: Set B contains the elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
  3. Combine the Elements: Now, let's combine these elements into a single set. We'll start by listing all the elements from set A, and then add any elements from set B that are not already in our list. This ensures we don't count any element twice.
  4. Form the Union (A ∪ B): The union of A and B will include all elements from both sets, without repetition. So, A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17}.
  5. Count the Elements: Finally, we count the number of elements in the union. By simply counting the elements in the set A ∪ B, we can determine the value of n(A ∪ B). This is the final step in our solution, where we quantify the result of our set operations.

Calculating n(A ∪ B)

Now that we have the union of sets A and B, which is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17}, we can easily find n(A ∪ B). This is just the number of elements in the set A ∪ B. Let's count them up!

We have the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, and 17. That's a total of 13 elements. So, n(A ∪ B) = 13. This is the final numerical answer to our problem, representing the cardinality of the union of the two sets. Understanding how to calculate the cardinality of set unions is crucial in many areas of mathematics and computer science, particularly in probability, statistics, and algorithm analysis.

The Formula Approach: Principle of Inclusion-Exclusion

There's another cool way to solve this using a formula called the Principle of Inclusion-Exclusion. This formula gives us a shortcut to finding the number of elements in the union of two sets. It's especially handy when dealing with larger sets where manually listing and counting elements might be tedious. The formula is:

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Where:

  • n(A) is the number of elements in set A.
  • n(B) is the number of elements in set B.
  • n(A ∩ B) is the number of elements in the intersection of A and B (i.e., the elements that are in both A and B).

Let's break down how to use this formula for our problem:

  1. Find n(A): Set A has 7 elements (2, 3, 5, 7, 11, 13, 17), so n(A) = 7.
  2. Find n(B): Set B has 10 elements (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), so n(B) = 10.
  3. Find A ∩ B: The intersection of A and B is the set of elements that are in both sets. Looking at our sets, we see that A ∩ B = {2, 3, 5, 7}. These are the elements that A and B have in common.
  4. Find n(A ∩ B): Since A ∩ B has 4 elements, n(A ∩ B) = 4.
  5. Apply the Formula: Now, we plug these values into the formula:
    • n(A ∪ B) = 7 + 10 - 4
    • n(A ∪ B) = 17 - 4
    • n(A ∪ B) = 13

So, using the Principle of Inclusion-Exclusion, we also find that n(A ∪ B) = 13. This method provides a more structured approach, especially useful when dealing with larger and more complex sets. It's a powerful tool in set theory and discrete mathematics, allowing us to calculate the size of unions without explicitly listing all the elements.

Answer

Therefore, the number of elements in the union of sets A and B, n(A ∪ B), is 13. This matches option B from the given choices. We arrived at this answer through two methods: first, by explicitly listing the elements in the union and counting them, and second, by using the Principle of Inclusion-Exclusion. Both methods confirm the same result, reinforcing our understanding of set operations and problem-solving techniques in mathematics.

Why This Answer is Correct

Option B, which is 13, is indeed the correct answer. We've demonstrated this through two different methods, which solidifies our confidence in the solution. The explicit listing method showed us exactly which elements are included in the union, and counting them directly gave us 13. The Principle of Inclusion-Exclusion provided a formulaic approach, which not only confirmed our previous result but also highlighted a powerful tool for set manipulation. This dual approach reinforces the accuracy of our answer and deepens our understanding of the underlying principles.

Common Mistakes to Avoid

When tackling problems like this, there are a few common pitfalls to watch out for. One frequent mistake is double-counting elements that appear in both sets. Remember, when forming the union, each unique element should only be counted once. Another error is omitting elements from either set. It's crucial to carefully consider all elements from both sets when forming the union. Lastly, a misunderstanding of the Principle of Inclusion-Exclusion can lead to incorrect calculations. Ensure you correctly identify the intersection of the sets and apply the formula accurately. Avoiding these common mistakes will help you solve similar problems with greater confidence and precision.

Conclusion

So, there you have it! We've successfully found that n(A ∪ B) = 13. We tackled this problem using two different methods: directly listing and counting the elements in the union, and applying the Principle of Inclusion-Exclusion. Both methods led us to the same answer, which is always a great sign that we're on the right track. Understanding set operations like unions and intersections is super important in math, and these skills can be applied in many real-world scenarios, from organizing data to solving logical puzzles. Keep practicing, and you'll become a set theory pro in no time! Remember, the key to mastering math is to break down complex problems into smaller, manageable steps, and always double-check your work to avoid those sneaky errors. Happy problem-solving, guys!