Draw the Hasse diagram for divisibility on the following set.

(𝑏) {1,2,3,5,7,11,13}

The Hasse diagram for divisibility illustrates how elements in a set divide one another, represented as a directed graph without loops. A line or edge from element \( a \) to element \( b \) means that \( a \) divides \( b \) and there is no other element in between them that divides \( b \).

Given the set \( \{1, 2, 3, 5, 7, 11, 13\} \), let's analyze how the elements divide each other:

1. \( 1 \) divides every other number (since 1 is a divisor of all integers).
2. None of the prime numbers divide each other (because prime numbers are divisible only by 1 and themselves).
3. Thus, the only element at the bottom of the Hasse diagram will be \( 1 \), and it will connect to each of the other elements (since 1 divides them all).
4. The other elements (2, 3, 5, 7, 11, 13) are all primes and are incomparable with each other in terms of divisibility.

### Steps to draw the diagram:

1. At the top, draw the element 1.
2. Draw lines from 1 to each of the prime numbers 2, 3, 5, 7, 11, and 13.
3. Since none of the prime numbers divide one another, there will be no lines between them.

The Hasse diagram looks like this:

```
        1
      / | \
     2  3  5  7  11  13
```

Each of the prime numbers (2, 3, 5, 7, 11, 13) is at the same level below 1, and there are no further connections among the primes.

Would you like to see a more detailed explanation of the steps?

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**Relative Questions:**
1. What is a Hasse diagram, and how is it used in other mathematical contexts?
2. Can you create a Hasse diagram for the set of divisors of 12?
3. How does the divisibility relation compare to other types of partial orders?
4. What are some other ways to visualize divisibility relations in number theory?
5. How does the concept of prime numbers affect the structure of Hasse diagrams for divisibility?

**Tip:** When drawing Hasse diagrams, always ensure there are no "unnecessary" edges between elements to keep the diagram minimal.

Draw the Hasse diagram for divisibility on the following set: {1,2,3,5,7,11,13}.

Explore how to draw the Hasse diagram for divisibility on the set {1, 2, 3, 5, 7, 11, 13}. This problem illustrates key concepts in divisibility and set theory, focusing on prime numbers and partial orders. Ideal for high school students in Grades 9-12.

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