Let's go through the answers for questions 1-10 from the image:

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### **1) Which of the following is an example of a reflexive relation?**
- A relation is **reflexive** if every element is related to itself, i.e., \( (a, a) \in R \) for all \( a \in A \).

**Options:**
- a) \( R = \{(1, 2), (2, 3)\} \) → This is not reflexive because neither \( (1, 1) \) nor \( (2, 2) \) nor \( (3, 3) \) are present.
- b) \( R = \{(1, 1), (2, 2)\} \) → This is reflexive because each element in the set is related to itself.
- c) \( R = \{(2, 3), (3, 1)\} \) → This is not reflexive.
- d) None of the above → Not correct because option b) is reflexive.

**Answer: b) \( R = \{(1, 1), (2, 2)\} \)**

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### **2) If a relation \( R \) is symmetric, which of the following must be true?**
- A relation is **symmetric** if \( (a, b) \in R \) implies \( (b, a) \in R \).

**Options:**
- a) \( (a,b) \in R \) implies \( (b,a) \in R \) → This is exactly the definition of a symmetric relation.
- b) \( (a,b) \in R \) implies \( (b,a) \notin R \) → This contradicts the definition of a symmetric relation.
- c) \( R \) is transitive → Symmetry does not imply transitivity.
- d) None of the above → Not true since option a) is correct.

**Answer: a) \( (a,b) \in R \) implies \( (b,a) \in R \)**

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### **3) The relation \( R = \{(x, y) \mid y = x\} \) is an example of which of the following?**
- This relation includes all pairs where \( x = y \), which means it contains all elements related to themselves.

**Options:**
- a) Reflexive and Symmetric → True, since \( x = y \) implies reflexivity and also symmetry.
- b) Reflexive, Symmetric, and Transitive → True, since the relation is also transitive (if \( x = y \) and \( y = z \), then \( x = z \)).
- c) Symmetric and Transitive → Not true because it is also reflexive.
- d) None of the above → Not correct because b) is true.

**Answer: b) Reflexive, Symmetric, and Transitive**

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### **4) Which of the following represents an antisymmetric relation?**
- A relation is **antisymmetric** if \( (a, b) \in R \) and \( (b, a) \in R \) implies \( a = b \).

**Options:**
- a) \( \{(1,2), (2,1)\} \) → This is not antisymmetric because \( (1, 2) \in R \) and \( (2, 1) \in R \), but \( 1 \neq 2 \).
- b) \( \{(1, 2), (2, 2)\} \) → This is antisymmetric because no contradiction arises.
- c) \( \{(1,1), (2,2)\} \) → This is antisymmetric because for all pairs, \( a = b \).
- d) None of the above → Not correct because both b) and c) are antisymmetric.

**Answer: c) \( \{(1,1), (2,2)\} \)**

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### **5) If a relation \( R \) on a set \( A \) is both reflexive and antisymmetric, which of the following is true?**
- Reflexive means all elements are related to themselves, and antisymmetric means no two different elements are mutually related unless they are equal.

**Options:**
- a) \( R \) is transitive → This is not necessarily true just because a relation is reflexive and antisymmetric.
- b) \( R \) is an equivalence relation → Not necessarily because it must also be symmetric.
- c) \( R \) must contain all ordered pairs of \( A \) → This is incorrect.
- d) None of the above → This is correct since none of the other options are true.



This page provides a detailed solution to questions 1-10 focusing on reflexive, symmetric, antisymmetric, and transitive relations. Explore core mathematical concepts and definitions related to relations and their properties, suitable for high school students in Grades 10-12.

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