Sets A And B: Finding Elements And Intersection

by SLV Team 48 views

Hey guys! Let's dive into this math problem involving sets A and B. We're going to figure out what elements are in set A and then find the intersection of sets A and B. It might sound a bit complicated, but we'll break it down step by step so it's super easy to understand. So, let's get started and explore the world of sets!

Determining Elements of Set A

Okay, so first, we need to figure out what's in set A. Remember, set A is defined as {x ∈ Z | -2√2 < x < 2√2}. This looks a bit math-y, but don't worry! Let's break it down. The key here is understanding what each part of this definition means. First off, x ∈ Z means that x is an integer—so we're talking about whole numbers (…-2, -1, 0, 1, 2…). Then we have the inequality -2√2 < x < 2√2. This tells us that x has to be greater than -2√2 and less than 2√2. Now, √2 is approximately 1.414. So, -2√2 is about -2.828, and 2√2 is about 2.828. Therefore, we're looking for integers that fall between -2.828 and 2.828. Think about the number line for a second. Which integers fit in this range? Well, that would be -2, -1, 0, 1, and 2. These are the whole numbers that are greater than -2.828 and less than 2.828. So, we've found our elements! We can now say that A = {-2, -1, 0, 1, 2}. See, it wasn't so bad once we took it piece by piece, right? The cool thing about understanding the elements of a set is that it's like laying the foundation for more complex operations. We know exactly what we're working with. When we move on to intersections and unions, having a clear picture of each set’s elements is super helpful. Plus, it's a great feeling when you can take a seemingly complicated definition and turn it into a simple list of numbers. This skill will come in handy as we tackle more challenging problems later on. It’s all about breaking things down and taking it one step at a time. So, now that we know set A, we're one step closer to cracking this problem wide open. Next up, we'll tackle set B and see how things get even more interesting!

Calculating A ∩ B

Alright, we've nailed down set A, so now it's time to tackle the second part of the problem: finding A ∩ B. First, let's recap what set B is. Set B is defined as {x ∈ Z | (12/(2x+1)) ∈ Z}. That means we’re looking for integers x such that when we plug them into the expression 12/(2x+1), the result is also an integer. In other words, 2x+1 has to be a divisor of 12. This might sound a little complicated, but we can break it down. What are the divisors of 12? Well, they're the numbers that divide evenly into 12, both positive and negative. So, we have ±1, ±2, ±3, ±4, ±6, and ±12. Now, we need to figure out which of these can be written in the form 2x+1. Remember, x has to be an integer. So, let's go through the divisors one by one and see if we can find a corresponding integer value for x.

  • If 2x+1 = 1, then 2x = 0, and x = 0. That's an integer! So, 0 is a possible element of B.
  • If 2x+1 = -1, then 2x = -2, and x = -1. Another integer! So, -1 is also in B.
  • If 2x+1 = 3, then 2x = 2, and x = 1. Yep, 1 is in B too.
  • If 2x+1 = -3, then 2x = -4, and x = -2. That works! -2 is in B.
  • If 2x+1 = 2, then 2x = 1, and x = 1/2. But that's not an integer, so 2 doesn't work.
  • If 2x+1 = -2, then 2x = -3, and x = -3/2. Again, not an integer, so -2 doesn't work.
  • If 2x+1 = 4, then 2x = 3, and x = 3/2. Not an integer.
  • If 2x+1 = -4, then 2x = -5, and x = -5/2. Not an integer.
  • If 2x+1 = 6, then 2x = 5, and x = 5/2. Not an integer.
  • If 2x+1 = -6, then 2x = -7, and x = -7/2. Not an integer.
  • If 2x+1 = 12, then 2x = 11, and x = 11/2. Not an integer.
  • If 2x+1 = -12, then 2x = -13, and x = -13/2. Not an integer.

So, the elements of set B are {-2, -1, 0, 1}. Now that we know both set A and set B, we can find their intersection, A ∩ B. The intersection of two sets is just the set of elements that are in both sets. We already determined that A = {-2, -1, 0, 1, 2} and B = {-2, -1, 0, 1}. Looking at these, we can see which elements they share. The elements that are in both A and B are -2, -1, 0, and 1. Therefore, A ∩ B = {-2, -1, 0, 1}.

Conclusion

And there you have it! We’ve successfully determined the elements of set A, figured out the elements of set B, and calculated their intersection. It might have seemed a bit daunting at first, but by breaking it down into smaller steps and thinking through each part, we were able to solve it. Remember, guys, in math (and in life!), tackling problems piece by piece often makes them much more manageable. Keep practicing, and you’ll be a set theory whiz in no time! Understanding the fundamentals like set operations, divisors, and inequalities is super important because they pop up in all sorts of mathematical contexts. So, pat yourselves on the back for tackling this problem, and keep up the great work! We've really flexed our math muscles today. Knowing how to define sets and then find relationships between them, like intersections, is a key skill. It builds a solid foundation for more advanced topics in mathematics and computer science. Plus, the ability to logically break down problems and work through them methodically is a skill that will serve you well in all areas of life. So, next time you encounter a math problem that looks a bit intimidating, just remember this example. Break it down, step by step, and you'll be surprised at what you can achieve! And remember, practice makes perfect. The more problems you work through, the more confident you'll become. Keep challenging yourselves, keep asking questions, and most importantly, keep having fun with math! You've got this!