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:

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

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!

Would you like details on any particular question, or have any questions so far?

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!