https://file.aiquickdraw.com/mathbot/file/1726758506012fixhof1m.jpg

I see you've uploaded an image of a set of math problems. Let me transcribe and provide guidance on them.

**7. Consider A = {1, 2, 3, 4, 5, 6}. Let R be the equivalence relation on A defined by:**
\[ R = \{ (1,1), (1,5), (2,2), (3,3), (4,4), (5,5), (6,6) \} \]
Find the partition of A induced by \( R \).

---

This problem is asking for the equivalence classes based on the relation \( R \). To solve, we examine the pairs and find how elements are grouped by equivalence.

### Explanation:
The relation \( R \) tells us which elements are related to each other, and equivalence relations induce partitions on sets.

Given:
- \( (1,1), (1,5), (5,5) \): Elements 1 and 5 are related, so they belong to the same equivalence class.
- \( (2,2) \): Element 2 is related only to itself, so it forms its own equivalence class.
- \( (3,3) \): Element 3 is related only to itself.
- \( (4,4) \): Element 4 is related only to itself.
- \( (6,6) \): Element 6 is related only to itself.

### Partition:
- \( \{1, 5\} \)
- \( \{2\} \)
- \( \{3\} \)
- \( \{4\} \)
- \( \{6\} \)

---

**8. Let S be the set of integers from 1 to 12. In S, define a relation as \( x \sim y \) if the product of the digits in \( x \) is the same as the product of the digits in \( y \). Find the number of members in the partition of S induced by this relation and list them.**

### Explanation:
Here we have a relation based on the product of the digits in each number in the set \( S = \{1, 2, 3, \dots, 12\} \).

1. For single-digit numbers, the product of digits is just the number itself:
   - \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \).

2. For two-digit numbers:
   - \( 10 \) has a product of \( 1 \times 0 = 0 \).
   - \( 11 \) has a product of \( 1 \times 1 = 1 \).
   - \( 12 \) has a product of \( 1 \times 2 = 2 \).

Now we group elements with the same product of digits:
- Product \( 0 \): \( \{10\} \).
- Product \( 1 \): \( \{1, 11\} \).
- Product \( 2 \): \( \{2, 12\} \).
- Product \( 3 \): \( \{3\} \).
- Product \( 4 \): \( \{4\} \).
- Product \( 5 \): \( \{5\} \).
- Product \( 6 \): \( \{6\} \).
- Product \( 7 \): \( \{7\} \).
- Product \( 8 \): \( \{8\} \).
- Product \( 9 \): \( \{9\} \).

### Number of partitions:
- \( 10 \) equivalence classes:
  \[
  \{10\}, \{1, 11\}, \{2, 12\}, \{3\}, \{4\}, \{5\}, \{6\}, \{7\}, \{8\}, \{9\}
  \]

---

Do you need details or further clarification on any part of the solution?

Here are 5 related questions to practice:
1. What is an equivalence relation, and how is it defined?
2. How do equivalence relations induce partitions in sets?
3. Given a set and an equivalence relation, how can you find the equivalence classes?
4. If you change the relation \( R \) in Problem 7, how does it affect the partition?
5. Can you think of other real-world situations where equivalence relations are useful?

**Tip**: When working with equivalence relations, always check for the properties of reflexivity, symmetry, and transitivity to confirm the relation!

Consider A = {1, 2, 3, 4, 5, 6}. Let R be the equivalence relation on A defined by: R = {(1,1), (1,5), (2,2), (3,3), (4,4), (5,5), (6,6)}. Find the partition of A induced by R. Additionally, let S be the set of integers from 1 to 12. In S, define a relation as x ~ y if the product of the digits in x is the same as the product of the digits in y. Find the number of members in the partition of S induced by this relation and list them.

Explore equivalence relations with set A = {1, 2, 3, 4, 5, 6} and find the partition induced by the given relation. Also, define a relation on S = {1, 2, ..., 12} based on the product of digits and determine the partition of S. This problem is ideal for high school students learning set theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation