Finding Sets: A, B, C, And D Explained With Easy Math

Hey guys! Let's dive into some cool math problems. We're going to figure out what's inside sets A, B, C, and D. Don't worry, it's not as scary as it sounds! We'll break down each problem step by step to make sure everyone understands. This is all about sets, divisors, and multiples – things that are super important in math. So, grab your pencils and let's get started. We'll be using basic math concepts to solve these problems, so even if you're not a math whiz, you should be able to follow along. This is like a fun puzzle that we can solve together. These are actually great exercises to boost your problem-solving skills, and who knows, maybe you'll even start enjoying math a little more!

Decoding Set A: Unveiling Natural Numbers and Divisors

Let's tackle Set A first! The definition says A = x | x ∈ N, x | 18 și 2x + 3 ≤ 15}. What does that even mean, right? Let's break it down piece by piece. First off, x ∈ N means that x is a natural number. Natural numbers are the counting numbers – 1, 2, 3, and so on. They don't include zero, fractions, or negative numbers. Next, x | 18 means that x divides 18, so x is a divisor of 18. This means that 18 divided by x gives you a whole number, with no remainders. And finally, 2x + 3 ≤ 15, which means we need to find values of x that, when we double them and add 3, give us a number less than or equal to 15. So, the question here is finding all natural numbers that divide 18 and also satisfy the inequality 2x + 3 ≤ 15. The first step involves finding all of the divisors of 18 1, 2, 3, 6, 9, and 18. Now, we have to check which of these divisors also satisfy the inequality. Let's test each one! If x = 1, then 2(1) + 3 = 5, which is less than 15. If x = 2, then 2(2) + 3 = 7, which is less than 15. If x = 3, then 2(3) + 3 = 9, which is less than 15. If x = 6, then 2(6) + 3 = 15, which is equal to 15. However, if x = 9, then 2(9) + 3 = 21, which is greater than 15, and if x = 18, then 2(18) + 3 = 39, which is also greater than 15. So, the values that work are 1, 2, 3, and 6. Therefore, the set A is {1, 2, 3, 6. This set contains all natural numbers that are divisors of 18 and make the inequality true. It's really just a process of elimination and checking to make sure our answers fit all the rules. It's like a math detective game! By breaking down the problem into smaller steps, it makes things so much easier to understand, right? The key is to take your time and not rush. Always double-check your work, and you'll do great! We're not just finding the answer; we're understanding why that's the answer.

Solving Set B: Navigating Divisors and Inequalities

Now, let’s move on to Set B. Set B = x | x ∈ N, x| 14 și 2x-1<19}. Similar to Set A, we need to find the values of x that fit the criteria. The first condition, x ∈ N, still means that x is a natural number. The second condition, x | 14, means that x divides 14, so it’s a divisor of 14. This means that 14 divided by x gives you a whole number with no remainders. The final condition, 2x - 1 < 19, means that we need to find the values of x that, when we double them and subtract 1, gives us a number less than 19. Let’s do the same thing and find all the divisors of 14 first. The divisors of 14 are 1, 2, 7, and 14. Now, we need to see which of these numbers work in the inequality 2x - 1 < 19. If x = 1, then 2(1) - 1 = 1, which is less than 19. If x = 2, then 2(2) - 1 = 3, which is less than 19. If x = 7, then 2(7) - 1 = 13, which is less than 19. Finally, if x = 14, then 2(14) - 1 = 27, which is not less than 19. Therefore, the values of x that fit both conditions are 1, 2, and 7. Thus, Set B is {1, 2, 7. See, it's pretty much the same method every time. The only thing that changes are the specific numbers and the inequality, but the logic stays the same. The key is to always break down the problem step by step, which helps with understanding the logic behind the solution. This process is very similar to how you solve problems in the real world. You break them down and then find ways to solve each piece of the puzzle. That's why math is so important! It teaches you how to think logically and solve complex problems in an easy way. Always remember that practice makes perfect, and the more problems you solve, the more comfortable you'll become with math concepts.

Cracking Set C: Exploring Divisors and Multiples

Alright, let's switch gears and work on Set C! Set C = x ∈ D10 şi x + 2 ∈ D12}. This one is a bit different. Here, D10 represents the set of divisors of 10, and D12 represents the set of divisors of 12. The first condition, x ∈ D10, means that x is a divisor of 10. The second condition, x + 2 ∈ D12, means that x + 2 must be a divisor of 12. Let's start with D10. The divisors of 10 are 1, 2, 5, and 10. So, we're looking for numbers from this list that, when you add 2 to them, give you a divisor of 12. Now we need to test each divisor of 10. Let's see what happens when we add 2 to each of them: If x = 1, then 1 + 2 = 3. Since 3 is a divisor of 12, then 1 works. If x = 2, then 2 + 2 = 4. Since 4 is a divisor of 12, then 2 works. If x = 5, then 5 + 2 = 7. However, 7 is not a divisor of 12. If x = 10, then 10 + 2 = 12. As 12 is a divisor of 12, then 10 works. Thus, the members of Set C are {1, 2, 10. Set C consists of the numbers that are divisors of 10, and when 2 is added to them, they become divisors of 12. This type of problem is all about understanding what the symbols and notation mean. Once you get a handle on that, solving these becomes easier. It might seem tricky at first, but with a little practice, you'll become a pro at these problems! We're not just finding answers; we're developing our math skills and critical thinking abilities. Remember to always double-check your work and to make sure your answers satisfy all the conditions given in the problem. The more you work on these types of problems, the more confident you'll become in your abilities.

Decoding Set D: Navigating Multiples and Divisors

Finally, let's explore Set D. Set D = {x ∈ M10 şi x-4 ∈ D26}. Here, M10 represents the set of multiples of 10, and D26 represents the set of divisors of 26. The first condition, x ∈ M10, means that x is a multiple of 10. The second condition, x - 4 ∈ D26, means that when you subtract 4 from x, you get a divisor of 26. Since multiples of 10 are numbers you get when you multiply 10 by any natural number (e.g. 10, 20, 30, 40, etc.), we need to test these multiples to see which ones fit the second condition. The divisors of 26 are 1, 2, 13, and 26. To find the numbers that fit, we need to add 4 to each of these divisors to see if the resulting number is a multiple of 10. If x - 4 = 1, then x = 5, but 5 is not a multiple of 10. If x - 4 = 2, then x = 6, but 6 is not a multiple of 10. If x - 4 = 13, then x = 17, but 17 is not a multiple of 10. If x - 4 = 26, then x = 30. And 30 is a multiple of 10. Thus, only one number satisfies both conditions, which is 30. Therefore, the set D is {30}. This type of problem highlights the relationships between multiples and divisors. It’s important to remember the difference between them, and how to identify them. Keep practicing, and don't be afraid to make mistakes. Mistakes are great learning opportunities. Keep up the great work, and congratulations on completing the problems! You've learned how to find the elements of different sets using divisors and multiples. Understanding these concepts helps you in algebra and other advanced areas of mathematics. With each problem, you're enhancing your analytical skills. So keep practicing and exploring the wonderful world of mathematics!

I hope that was a helpful explanation, guys! Let me know if you have any questions or if you want to explore more math problems. Keep up the fantastic work! Remember, it's all about practice and having fun with it.