Listing Sets: Natural, Whole, Factors, And Primes

by Tim Redaksi 50 views
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Let's break down how to represent sets using two common methods: listing elements and set-builder notation. We'll tackle natural numbers, whole numbers, factors, and prime numbers. Get ready to dive into the world of set theory!

1. Set B: Natural Numbers Between 1 and 20

Natural numbers, guys, are the counting numbers – 1, 2, 3, and so on. This question wants us to list all the natural numbers between 1 and 20, meaning we don't include 1 or 20 themselves. So, let's get to it!

Listing Elements

When listing elements, we simply write out all the members of the set, separated by commas, and enclosed in curly braces {}. For set B, this looks like:

B = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

Easy peasy, right? We've just listed every natural number that fits the criteria. Make sure you don't miss any numbers in between! Double-checking is always a good idea.

Set-Builder Notation

Now, let's represent the same set using set-builder notation. This is a more concise way of defining a set based on a specific rule or condition. The general form is:

{x | condition(x)}

This reads as "the set of all x such that condition(x) is true". For set B, we can write:

B = {x | x ∈ ℕ, 1 < x < 20}

Let's break this down:

  • x ∈ ℕ: This means "x is an element of the set of natural numbers" (ℕ is the symbol for natural numbers).
  • 1 < x < 20: This means "x is greater than 1 and less than 20".

So, the entire expression reads as "the set of all x such that x is a natural number, and x is greater than 1 and less than 20". This is just a fancy way of saying the same thing we said earlier, but it's much more compact, especially when dealing with more complex sets!

Understanding both methods is crucial. Listing is straightforward for smaller sets, while set-builder notation shines when defining sets with more intricate rules or infinite members.

2. Set P: Whole Numbers Less Than 20

Time to tackle whole numbers! Remember, whole numbers are similar to natural numbers, but they include zero. So, the set of whole numbers is 0, 1, 2, 3, and so on. This time, we want all the whole numbers less than 20. Let's see how to represent this set in both ways.

Listing Elements

Listing the elements of set P is pretty straightforward. We start with 0 and go up to 19, since we want numbers less than 20:

P = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

Just like before, we enclose all the elements within curly braces and separate them with commas. Make sure you include 0, since it's a whole number, and that you stop at 19, because 20 is not less than 20.

Set-Builder Notation

Now, for the set-builder notation. We'll use a similar approach to the previous example, but with a slight tweak to account for whole numbers. We can write:

P = {x | x ∈ 𝕎, x < 20}

Here's what each part means:

  • x ∈ 𝕎: This means "x is an element of the set of whole numbers" (𝕎 is the symbol for whole numbers).
  • x < 20: This means "x is less than 20".

So, the whole expression reads as "the set of all x such that x is a whole number, and x is less than 20". Again, this is a concise way to define the set based on a specific condition. As you can see, set-builder notation is powerful for defining sets based on rules, even when the sets are infinite.

3. Set G: Factors of 12

Let's switch gears to factors. A factor of a number is an integer that divides evenly into that number. For example, the factors of 6 are 1, 2, 3, and 6. So, for set G, we need to find all the factors of 12.

Listing Elements

To list the elements, let's systematically find all the numbers that divide 12 without leaving a remainder:

  • 1 divides 12 (12 / 1 = 12)
  • 2 divides 12 (12 / 2 = 6)
  • 3 divides 12 (12 / 3 = 4)
  • 4 divides 12 (12 / 4 = 3)
  • 6 divides 12 (12 / 6 = 2)
  • 12 divides 12 (12 / 12 = 1)

So, the set of factors of 12 is:

G = {1, 2, 3, 4, 6, 12}

Make sure you include both 1 and the number itself (in this case, 12) when listing factors. A good strategy is to start with 1 and work your way up, checking each number to see if it divides evenly. If it does, add it to your list!

Set-Builder Notation

Representing this with set-builder notation is a little different. We need to express the condition that x is a factor of 12. We can write:

G = {x | x ∈ ℤ, x divides 12}

Let's break it down:

  • x ∈ ℤ: This means "x is an element of the set of integers" (ℤ is the symbol for integers, which include positive and negative whole numbers, and zero).
  • x divides 12: This means "12 is divisible by x" or "x is a factor of 12".

So, the entire expression reads as "the set of all x such that x is an integer, and x divides 12". It's important to specify that x is an integer because factors are always integers.

4. Set J: Prime Numbers Less Than or Equal to 13

Last but not least, let's tackle prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few prime numbers are 2, 3, 5, 7, and so on. We want to find all the prime numbers less than or equal to 13.

Listing Elements

Let's identify the prime numbers that meet our criteria:

  • 2 is prime (only divisible by 1 and 2)
  • 3 is prime (only divisible by 1 and 3)
  • 5 is prime (only divisible by 1 and 5)
  • 7 is prime (only divisible by 1 and 7)
  • 11 is prime (only divisible by 1 and 11)
  • 13 is prime (only divisible by 1 and 13)

So, the set of prime numbers less than or equal to 13 is:

J = {2, 3, 5, 7, 11, 13}

Remember that 1 is not a prime number. It's a common mistake to include it, so be careful!

Set-Builder Notation

Using set-builder notation, we can write:

J = {x | x ∈ ℕ, x is prime, x ≤ 13}

Here's what it means:

  • x ∈ ℕ: This means "x is an element of the set of natural numbers".
  • x is prime: This means "x is a prime number".
  • x ≤ 13: This means "x is less than or equal to 13".

Therefore, the entire expression reads as "the set of all x such that x is a natural number, x is prime, and x is less than or equal to 13". This concisely defines the set using the properties of prime numbers.

By working through these examples, you've gained a solid understanding of how to represent sets using both listing elements and set-builder notation. Keep practicing, and you'll become a set theory master in no time! Remember the definitions of each type of number and you will solve the questions with ease. Good luck!