https://file.aiquickdraw.com/mathbot/file/1727088665551b8q2zoko.jpg

It seems like you've uploaded an image containing a set of mathematical problems and relational questions. I’ll walk through one of the questions based on the parts I can see clearly.

**22a (ii)** appears to ask you to determine whether each of the following relations on the set \( \{1, 2, 3, 4\} \) is reflexive, symmetric, antisymmetric, and transitive. The relations listed are:

1. \( R_1 = \{(2, 1), (3, 2), (3, 3)\} \)
2. \( R_2 = \{(1, 1), (2, 2), (3, 3), (4, 4)\} \)
3. \( R_3 = \{(1, 2), (2, 1), (2, 3), (3, 4)\} \)
4. \( R_4 = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3), (3, 4)\} \)
5. \( R_5 = \{(1, 3), (2, 2), (3, 3), (4, 4)\} \)

Let's analyze one of these relations as an example.

### Example: \( R_1 = \{(2, 1), (3, 2), (3, 3)\} \)
1. **Reflexive**: For a relation to be reflexive, every element in the set must relate to itself. The set is \( \{1, 2, 3, 4\} \), so we need the pairs \( (1, 1), (2, 2), (3, 3), (4, 4) \). Since \( (2, 2) \) and \( (1, 1) \) are missing, it is **not reflexive**.

2. **Symmetric**: A relation is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \). In \( R_1 \), we have \( (2, 1) \) but not \( (1, 2) \), so it is **not symmetric**.

3. **Antisymmetric**: A relation is antisymmetric if for any \( (a, b) \in R \) and \( (b, a) \in R \), then \( a = b \). There is no pair such that both \( (a, b) \) and \( (b, a) \) appear in \( R_1 \), so the relation is **antisymmetric**.

4. **Transitive**: A relation is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \). There’s no situation where we can verify a transitive property in \( R_1 \), so it is **transitive**.

Would you like me to analyze the remaining relations or assist with another part of the image?

Here are 5 related questions for practice:
1. How do you prove if a relation is an equivalence relation?
2. What is the difference between symmetric and antisymmetric relations?
3. Can a relation be both symmetric and antisymmetric? If yes, under what conditions?
4. How would you check if a relation is a partial order?
5. How does the concept of transitivity apply in real-world relations like "is a parent of"?

**Tip**: When determining whether a relation is reflexive, always check if every element relates to itself (i.e., \( (a, a) \) should be in the set for all elements \( a \)).

For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, symmetric, antisymmetric, and transitive.

Learn how to analyze relations on the set {1, 2, 3, 4} to determine if they are reflexive, symmetric, antisymmetric, and transitive. This problem involves set theory and the properties of relations, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation