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 which is reflexive, transitive but not symmetric:
- Reflexive: A relation \( R \) is reflexive if for all \( a \in A \), \( (a, a) \in R \).
- Transitive: A relation \( R \) is transitive if for all \( (a, b) \in R \) and \( (b, c) \in R \), \( (a, c) \in R \).
- Symmetric: A relation \( R \) is symmetric if for all \( (a, b) \in R \), \( (b, a) \in R \).
Now, let's check each relation for reflexivity, transitivity, and symmetry.
- \( R_1 = \{(2, 3), (3, 2)\} \):
- Not reflexive (missing \( (1, 1) \), \( (2, 2) \), \( (3, 3) \)).
- Not transitive because we don't have \( (2, 2) \) or \( (3, 3) \) for transitivity.
- Not symmetric since \( (2, 3) \) is in \( R_1 \), but \( (3, 2) \) is not.
- \( R_2 = \{(1, 2), (1, 3), (3, 2)\} \):
- Not reflexive (missing \( (2, 2) \), \( (3, 3) \)).
- Transitive: It is transitive since if we have \( (1, 2) \) and \( (2, 3) \), we also have \( (1, 3) \), and similarly for other combinations.
- Not symmetric, because \( (1, 2) \) is in \( R_2 \), but \( (2, 1) \) is not.
- \( R_3 = \{(1, 2), (2, 1), (1, 1)\} \):
- Reflexive because it includes \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \).
- Symmetric because for every pair \( (a, b) \), \( (b, a) \) is also present.
- Not transitive because there is no \( (1, 3) \), which makes it not transitive.
Thus, \( R_3 \) is reflexive and symmetric but not transitive.
(ii) Identify the relation which is reflexive and symmetric but not transitive:
From the analysis above, \( R_3 \) is reflexive and symmetric but not transitive.
(iii) Identify the relations which are symmetric but neither reflexive nor transitive:
- \( R_1 \) is symmetric but neither reflexive nor transitive.
- \( R_5 \) 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, it needs to be reflexive, symmetric, and transitive. - Reflexive: We need to add \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \). - Symmetric: Add \( (2, 1) \) and \( (3, 1) \) (since \( (1, 2) \), \( (1, 3) \) are already present, but their reverse pairs are missing). - Transitive: Ensure the necessary transitive pairs are present, which may be covered after adding the missing symmetric pairs. Thus, the pairs to be added are: \[ (1, 1), (2, 2), (3, 3), (2, 1), (3, 1). \]