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The image contains a series of questions related to set theory, relations, and graph theory. Here's a breakdown of the sections:

### Problems [1-5]:
These are about binary relations and graph representations:
1. Asking to define "binary relation" on the set \( A \times B \).
2. Regarding reflexivity and graphical representation.
3. Introduces sets \( A, B, C \) and examines conditions involving transitivity in the relations.
4. Requests describing the relation \( R \subseteq A \times A \) and its directed graph.
5. Definition of an "initial vertex" in directed graphs.

### Problems [6-17]:
These questions ask for determining reflexive, symmetric, antisymmetric, and transitive properties for various relations:
- Problems [6-9] involve determining the properties of relations on set \( A = \{1, 2, 3, 4, 5\} \).
- Problems [10-13] extend this to the relation \( R = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} | x = 1 \text{ or } y = 1\} \).
- Problems [14-17] relate to determining properties for a directed graph based on set \( A = \{a, b, c, d\} \), with an accompanying graph.

### Problems [18-20]:
- Question 18 asks about relation \( R = \{(a, b) | ab = 1 \mod 1000\} \).
- Question 19 introduces matrix \( M_R \) to describe the relation \( R^4 \).
- Question 20 asks about the distinct non-reflexive relations on a set of size 16.

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I can help you solve these problems step by step. Do you have a specific question in mind, or would you like me to start solving from the beginning? Let me know if you need a detailed solution or explanation for any of the sections! 

Here are five questions expanding from this:

1. What is the definition of a binary relation on sets?
2. How does one determine if a relation is reflexive, symmetric, or transitive?
3. What are the properties of transitive closure in relation to directed graphs?
4. Can you explain the concept of equivalence relations and their properties?
5. How are adjacency matrices used to represent directed graphs?

**Tip**: When working with relations, understanding their properties (like reflexivity, symmetry, transitivity) helps in interpreting their graph representations and matrix forms.

The user submitted a series of mathematical problems related to set theory, relations, and graph theory, including determining properties like reflexivity, symmetry, antisymmetry, and transitivity for various relations, and exploring adjacency matrices and directed graphs.

This webpage explains key concepts in set theory and relations, including binary relations, reflexivity, symmetry, transitivity, and their representation using directed graphs and adjacency matrices. Learn how to solve complex problems involving these mathematical structures.

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