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The image contains multiple-choice questions on relations in set theory and properties like reflexivity, symmetry, antisymmetry, and transitivity. I will transcribe the questions so we can go over them one by one.

Here are the questions:

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1) **Which of the following is an example of a reflexive relation?**
   - a) \( R = \{(1,2), (2,3)\} \)
   - b) \( R = \{(1,1), (2,2)\} \)
   - c) \( R = \{(2,3), (3,1)\} \)
   - d) None of the above

2) **If a relation \( R \) is symmetric, which of the following must be true?**
   - a) \( (a,b) \in R \) implies \( (b,a) \in R \)
   - b) \( (a,b) \in R \) implies \( (b,a) \notin R \)
   - c) \( R \) is transitive
   - d) None of the above

3) **The relation \( R = \{(x,y) \mid y = x\} \) is an example of which of the following?**
   - a) Reflexive and Symmetric
   - b) Reflexive, Symmetric, and Transitive
   - c) Symmetric and Transitive
   - d) None of the above

4) **Which of the following represents an antisymmetric relation?**
   - a) \( \{(1,2), (2,1)\} \)
   - b) \( \{(1,2), (2,2)\} \)
   - c) \( \{(1,1), (2,2)\} \)
   - d) None of the above

5) **If a relation \( R \) on a set \( A \) is both reflexive and antisymmetric, which of the following is true?**
   - a) \( R \) is transitive
   - b) \( R \) is an equivalence relation
   - c) \( R \) must contain all ordered pairs of \( A \)
   - d) None of the above

6) **How many relations can be defined on a set \( A \) with \( |A| = 3 \)?**
   - a) 8
   - b) 16
   - c) 512
   - d) 64

7) **For a relation \( R \) on set \( A \), what is the cardinality of \( A \times A \) when \( |A| = 4 \)?**
   - a) 8
   - b) 16
   - c) 16
   - d) 32

8) **Which of the following properties does every equivalence relation satisfy?**
   - a) Reflexive only
   - b) Symmetric only
   - c) Reflexive, Symmetric, and Transitive
   - d) Antisymmetric

9) **Which of the following is a valid example of an equivalence relation?**
   - a) \( R = \{(1,2), (2,1)\} \)
   - b) \( R = \{(1,1), (2,2)\} \)
   - c) \( R = \{(1,3), (2,2)\} \)
   - d) \( R = \{(2,1), (3,1)\} \)

10) **In terms of sets, the transitive closure of a relation \( R \) is defined as:**
    - a) The smallest transitive relation that contains \( R \)
    - b) The largest transitive relation that contains \( R \)
    - c) The smallest reflexive relation that contains \( R \)
    - d) None of the above

11) **The relation \( R \) on set \( A = \{1, 2, 3\} \) is given by \( R = \{(1,1), (2,2), (3,3)\} \). This relation is:**
    - a) Reflexive
    - b) Symmetric
    - c) Transitive
    - d) All of the above

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If you'd like, I can guide you through solving these questions or explain any specific ones. Let me know where you want to begin!

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### Relative Questions
1. What does it mean for a relation to be reflexive?
2. Can a relation be symmetric but not transitive? Why or why not?
3. How can you determine whether a relation is antisymmetric?
4. What are the key properties of an equivalence relation?
5. How is the cardinality of a relation on a set calculated?

### Tip:
Always remember that equivalence relations must satisfy reflexivity, symmetry, and transitivity simultaneously!

Multiple-choice questions on relations in set theory, including concepts like reflexive, symmetric, antisymmetric, and transitive relations.

Solve multiple-choice questions on relations in set theory, covering reflexive, symmetric, antisymmetric, and transitive properties. These questions are ideal for students studying discrete mathematics at the college level.

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