Sets By Extension: Prime Numbers, Divisors, Multiples

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Hey guys! Today, we're going to break down how to define sets by extension. This means listing out all the elements that belong to each set. We'll tackle prime numbers, divisors, multiples, and a couple of simple equations. Let's dive right in!

Set D: Prime Numbers Greater Than 1 and Less Than or Equal to 11

Okay, so our first task is to determine the set D, where D = {x | x is a prime number and 1 < x ≤ 11}. Prime numbers are whole numbers greater than 1 that have only two divisors: 1 and themselves. We need to find all the prime numbers that are greater than 1 and less than or equal to 11. Let's list them out:

  • 2 is a prime number because its only divisors are 1 and 2.
  • 3 is a prime number because its only divisors are 1 and 3.
  • 5 is a prime number because its only divisors are 1 and 5.
  • 7 is a prime number because its only divisors are 1 and 7.
  • 11 is a prime number because its only divisors are 1 and 11.

So, putting it all together, the set D defined by extension is:

D = {2, 3, 5, 7, 11}

Remember, defining a set by extension simply means writing out each element within curly braces and separating them by commas. Make sure to only include elements that actually satisfy the given condition. This is a foundational concept in set theory, and getting comfortable with it will really help you in more advanced math courses. Keep practicing, and you'll become a pro in no time!

Set F: Divisors of 8

Next up, we're figuring out set F, where F = {x | x is a divisor of 8}. Divisors are numbers that divide evenly into another number. In this case, we want all the numbers that divide evenly into 8. Let's find them. Think of it as what numbers you can divide 8 by and get a whole number result. Start from 1 and go up.

  • 1 is a divisor of 8 because 8 ÷ 1 = 8.
  • 2 is a divisor of 8 because 8 ÷ 2 = 4.
  • 4 is a divisor of 8 because 8 ÷ 4 = 2.
  • 8 is a divisor of 8 because 8 ÷ 8 = 1.

So, the set F, defined by extension, looks like this:

F = {1, 2, 4, 8}

When finding divisors, always remember to include 1 and the number itself, as they are always divisors. This set is relatively straightforward, but understanding divisors is crucial for many other math topics, like prime factorization and greatest common divisors. Keep an eye out for those, and you'll see how these concepts build upon each other. Remember to check if the question has further contraints, such as being only positive numbers.

Set G: Multiples of 3

Now, let's tackle set G, where G = {x | x is a multiple of 3}. Multiples are the results you get when you multiply a number by an integer. In this case, we need all the multiples of 3. Since there's no upper limit given, this set will be infinite. We can list the first few elements to show the pattern:

  • 3 x 1 = 3
  • 3 x 2 = 6
  • 3 x 3 = 9
  • 3 x 4 = 12
  • 3 x 5 = 15

And so on. Therefore, we can represent the set G by extension as:

G = {3, 6, 9, 12, 15, ...}

The ellipsis (...) indicates that the set continues infinitely. Identifying multiples is incredibly important, especially when you're dealing with sequences, series, and modular arithmetic. It’s one of those fundamental skills that keeps popping up, so make sure you're comfortable with it. Some questions might have conditions such as natural numbers or whole numbers, make sure to read the questions well.

Set H: Natural Numbers Satisfying x + 5 = 8

Alright, time for set H, where H = {x | x belongs to natural numbers; x + 5 = 8}. Natural numbers are positive whole numbers (1, 2, 3, ...). We need to find the natural number x that satisfies the equation x + 5 = 8. To solve for x, we subtract 5 from both sides of the equation:

x + 5 - 5 = 8 - 5 x = 3

Since 3 is a natural number, it belongs to the set H. Therefore, the set H defined by extension is:

H = {3}

This set only contains one element. Solving simple equations like this is a core skill in algebra. Understanding the properties of different number sets (like natural numbers, integers, and real numbers) is also very important for correctly interpreting and solving mathematical problems. Keep practicing these algebraic manipulations, and you'll be solving more complex equations in no time. Always pay attention to what number set the solutions have to belong to.

Set E: Integers Satisfying 2x + 1 = 5

Last but not least, we're tackling set E, where E = {x | x belongs to integers; 2x + 1 = 5}. Integers are whole numbers, including positive numbers, negative numbers, and zero (... -2, -1, 0, 1, 2 ...). We need to find the integer x that satisfies the equation 2x + 1 = 5. Let's solve for x:

First, subtract 1 from both sides:

2x + 1 - 1 = 5 - 1 2x = 4

Now, divide both sides by 2:

2x / 2 = 4 / 2 x = 2

Since 2 is an integer, it belongs to the set E. So, the set E defined by extension is:

E = {2}

Just like set H, this set also contains only one element. Again, the ability to solve these types of equations is super important. And remember, integers include negative numbers and zero, so always check if your solution fits within the correct number set.

Final Thoughts

So, there you have it! We've successfully determined the elements of sets D, F, G, H, and E by extension. Remember, defining a set by extension means listing out all the elements that satisfy the given conditions. Whether it's prime numbers, divisors, multiples, or solutions to equations, the process is the same. Keep practicing, and you'll become a set theory whiz in no time! Keep an eye out for extra conditions!