Set Operations: Intersection And Union Explained

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Hey guys! Let's dive into some cool math problems involving sets. We're going to explore how to find the intersection and union of sets, specifically with real numbers, and then see how those results interact with natural numbers and integers. It might sound a bit intimidating at first, but I promise it's pretty straightforward once you get the hang of it. We'll break down the steps, explain the concepts, and hopefully make it all crystal clear. So, grab your pencils (or your favorite coding environment), and let's get started!

Understanding the Basics: Sets, Intervals, and Numbers

First off, let's make sure we're all on the same page with the basic definitions. A set is simply a collection of distinct objects. In our case, these objects will be numbers. We'll be working with sets of real numbers (denoted by R), which include all rational and irrational numbers. The problems introduce two sets, A and B, defined using inequalities. Inequalities help us define intervals of numbers. An interval is a set of real numbers that lie between two given numbers. These intervals can be open (not including the endpoints), closed (including the endpoints), or half-open (including one endpoint but not the other). We'll also need to know about some specific types of numbers:

  • Natural Numbers (N): These are the counting numbers: 1, 2, 3, and so on. They are positive whole numbers.
  • Integers (Z): These include all whole numbers (both positive and negative), including zero: ..., -3, -2, -1, 0, 1, 2, 3, ...

Now, the main operations we're interested in are intersection and union.

  • Intersection (∩): The intersection of two sets is a new set that contains only the elements that are common to both sets. Think of it as the overlap between the sets. If two sets have no elements in common, their intersection is an empty set (denoted by ∅).
  • Union (∪): The union of two sets is a new set that contains all the elements from both sets, without any repetitions. It's like combining the two sets into one big set.

The Set A and its interval

Let's analyze the set A defined as A = {x ∈ R | -4 ≤ (5x+7)/2 < 16}. This means A contains all real numbers x that satisfy the inequality -4 ≤ (5x+7)/2 < 16. To find the interval for A, we need to solve this compound inequality. We do this by applying operations to all parts of the inequality simultaneously. First, we'll multiply all parts of the inequality by 2:

-4 * 2 ≤ 5x + 7 < 16 * 2 -8 ≤ 5x + 7 < 32

Next, we subtract 7 from all parts of the inequality:

-8 - 7 ≤ 5x < 32 - 7 -15 ≤ 5x < 25

Finally, we divide all parts of the inequality by 5:

-15/5 ≤ x < 25/5 -3 ≤ x < 5

So, set A consists of all real numbers x such that -3 ≤ x < 5. In interval notation, we can write this as A = [-3, 5). This means A includes -3 but does not include 5. Any number that fits the condition will be included in the interval.

The Set B and its Interval

Now, let's look at set B, defined as B = {x ∈ R | -1 ≤ (7x+12)/9 ≤ 6}. Similar to set A, this means B contains all real numbers x that satisfy the inequality -1 ≤ (7x+12)/9 ≤ 6. To find the interval for B, we solve this compound inequality. First, we multiply all parts of the inequality by 9:

-1 * 9 ≤ 7x + 12 ≤ 6 * 9 -9 ≤ 7x + 12 ≤ 54

Next, we subtract 12 from all parts of the inequality:

-9 - 12 ≤ 7x ≤ 54 - 12 -21 ≤ 7x ≤ 42

Finally, we divide all parts of the inequality by 7:

-21/7 ≤ x ≤ 42/7 -3 ≤ x ≤ 6

So, set B consists of all real numbers x such that -3 ≤ x ≤ 6. In interval notation, we can write this as B = [-3, 6]. This means B includes both -3 and 6. Any number that fits the condition will be included in the interval.

Calculating the Intersection of A, B with Natural Numbers (A ∩ B ∩ N)

Alright, now that we know what sets A and B look like, let's find their intersection. The intersection of A and B (denoted as A ∩ B) is the set of all elements that belong to both A and B. From the previous calculations, we know:

  • A = [-3, 5)
  • B = [-3, 6]

The intersection, A ∩ B, includes all the numbers that are in both the intervals of A and B. Since A includes numbers from -3 up to (but not including) 5, and B includes numbers from -3 up to and including 6, the intersection will start at -3 (inclusive) and go up to 5 (exclusive) because 5 is not included in A. Thus A ∩ B = [-3, 5). Now we need to find the intersection of A ∩ B and the set of natural numbers (N). Remember, natural numbers are the positive whole numbers (1, 2, 3, ...). The intersection A ∩ B ∩ N will include only the natural numbers that are also in the interval [-3, 5). These numbers are 1, 2, 3, and 4. So, A ∩ B ∩ N = {1, 2, 3, 4}.

Step-by-Step Breakdown:

  1. Find A ∩ B: A = [-3, 5), B = [-3, 6] => A ∩ B = [-3, 5).
  2. Identify Natural Numbers in A ∩ B: Natural numbers in the interval [-3, 5) are 1, 2, 3, 4.
  3. Result: A ∩ B ∩ N = {1, 2, 3, 4}.

Calculating the Union of A, B with Integers (A ∪ B ∩ Z)

Now, let's compute the union. The union of A and B (denoted as A ∪ B) is the set of all elements that belong to either A or B (or both). We know that A = [-3, 5) and B = [-3, 6]. The union A ∪ B will include all numbers from the interval of A combined with all the numbers from the interval of B. This means it will start at -3 (inclusive, since it's in both A and B), and go all the way up to 6 (inclusive, since it's in B). Thus, A ∪ B = [-3, 6]. Next, we need to find the intersection of A ∪ B and the set of integers (Z). Remember, integers are all whole numbers (positive, negative, and zero). The intersection A ∪ B ∩ Z will include only the integers that are also in the interval [-3, 6]. These numbers are -3, -2, -1, 0, 1, 2, 3, 4, 5, and 6. So, A ∪ B ∩ Z = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.

Step-by-Step Breakdown:

  1. Find A ∪ B: A = [-3, 5), B = [-3, 6] => A ∪ B = [-3, 6].
  2. Identify Integers in A ∪ B: Integers in the interval [-3, 6] are -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.
  3. Result: A ∪ B ∩ Z = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.

Conclusion: Summary of Results

In summary, we've successfully computed the required set operations. Here's a recap:

  • A ∩ B ∩ N = {1, 2, 3, 4}: This set includes the natural numbers that are in the intersection of sets A and B.
  • A ∪ B ∩ Z = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}: This set includes all the integers that are in the union of sets A and B.

I hope that was helpful and that you now have a better understanding of how to work with sets, intervals, intersections, unions, and different types of numbers. These are fundamental concepts in mathematics and are used widely across various fields. Keep practicing, and you'll become a pro in no time! If you have any questions, feel free to ask. And don't forget to always double-check your work, especially when dealing with inequalities and interval endpoints! Keep practicing and expanding your understanding of set theory, and you'll find it an invaluable tool for problem-solving in mathematics and beyond. Great job! Keep exploring!