Question

A classroom teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set \( A = \{1, 2, 3\} \): \[ R_1 = \{(2, 3), (3, 2)\}, \quad R_2 = \{(1, 2), (1, 3), (3, 2)\}, \quad R_3 = \{(1, 2), (2, 1), (1, 1)\}, \] \[ R_4 = \{(1, 1), (1, 2), (3, 3), (2, 2)\}, \quad R_5 = \{(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)\}. \] The students are asked to answer the following questions about the above relations:
(i) Identify the relation which is reflexive, transitive but not symmetric.
\ (ii) Identify the relation which is reflexive and symmetric but not transitive.
(iii) Identify the relations which are symmetric but neither reflexive nor transitive.
OR
(iii) What pairs should be added to the relation \( R_2 \) to make it an equivalence relation?

Hint:

For a relation to be an equivalence relation, it must be reflexive, symmetric, and transitive. Ensure all conditions are satisfied by adding the necessary pairs for reflexivity, symmetry, and transitivity.

Answer 1

(i) Identify the relation that is reflexive, transitive, but not symmetric:
- Reflexive: A relation \( R \) on a set \( A \) is reflexive if \( (a, a) \in R \) for all \( a \in A \).
- Transitive: A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).
- Symmetric: A relation \( R \) is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \).
Analyzing the given relations:
- \( R_1 = \{(2, 3), (3, 2)\} \):
- Not reflexive (missing \( (1, 1) \), \( (2, 2) \), \( (3, 3) \)).
- Not transitive (e.g., \( (2, 3) \in R_1 \) and \( (3, 2) \in R_1 \), but \( (2, 2) otin R_1 \)).
- Symmetric (since \( (2, 3) \in R_1 \) implies \( (3, 2) \in R_1 \), and vice-versa).
- \( R_2 = \{(1, 2), (1, 3), (3, 2)\} \):
- Not reflexive (missing \( (2, 2) \), \( (3, 3) \)).
- Transitive (if \( (1, 2) \in R_2 \) and \( (2, 3) \in R_2 \), then \( (1, 3) \in R_2 \). All other applicable pairs satisfy transitivity).
- Not symmetric ( \( (1, 2) \in R_2 \) but \( (2, 1) otin R_2 \)).
- \( R_3 = \{(1, 2), (2, 1), (1, 1)\} \):
- Not reflexive (missing \( (2, 2) \), \( (3, 3) \)).
- Symmetric ( \( (1, 2) \in R_3 \) implies \( (2, 1) \in R_3 \); \( (1, 1) \in R_3 \) implies \( (1, 1) \in R_3 \)).
- Not transitive ( \( (2, 1) \in R_3 \) and \( (1, 2) \in R_3 \), but \( (2, 2) otin R_3 \)).
The relation \( R_2 \) is transitive and not symmetric. It is also not reflexive on a set \( A = \{1, 2, 3\} \) because \( (2, 2) \) and \( (3, 3) \) are missing. If we assume the set is \( A = \{1, 2, 3\} \), then \( R_2 \) is not reflexive. However, if we need to find a relation that is reflexive, transitive, but not symmetric, none of the provided examples strictly fit without further assumptions or modifications to the implied universal set. (ii) Identify the relation that is reflexive and symmetric but not transitive:
None of the provided relations \( R_1, R_2, R_3 \) are reflexive and symmetric but not transitive based on a universal set \( A=\{1, 2, 3\} \). The analysis for \( R_3 \) incorrectly stated it was reflexive. (iii) Identify the relations that are symmetric but neither reflexive nor transitive:
- \( R_1 \) is symmetric but neither reflexive nor transitive.
- \( R_3 \) is symmetric but neither reflexive nor transitive.
OR (iii) What pairs should be added to the relation \( R_2 \) to make it an equivalence relation? To make \( R_2 = \{(1, 2), (1, 3), (3, 2)\} \) an equivalence relation on \( A = \{1, 2, 3\} \), it must be reflexive, symmetric, and transitive.
- Reflexive: We need to add \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \).
- Symmetric: Since \( (1, 2) \in R_2 \), we need \( (2, 1) \in R_2 \). Since \( (1, 3) \in R_2 \), we need \( (3, 1) \in R_2 \). \( (3, 2) \in R_2 \) implies \( (2, 3) \in R_2 \).
- Transitive: After adding the required reflexive and symmetric pairs, we must ensure transitivity. For example, \( (1, 2) \in R_2 \) and \( (2, 3) \in R_2 \) implies \( (1, 3) \in R_2 \) (which is already present). \( (1, 3) \in R_2 \) and \( (3, 2) \in R_2 \) implies \( (1, 2) \in R_2 \) (which is already present).
The pairs to be added are: \[ (1, 1), (2, 2), (3, 3), (2, 1), (3, 1), (2, 3). \]