Question

A relation \(R\) on \[ A=\{1,2,3\} \] is defined as \[ R=\{(1,1),(3,3),(1,2)\}. \] Is \(R\) a symmetric relation? Justify your answer.

Write the smallest relation set \(R_1\) such that \[ R\cup R_1 \] becomes an equivalence relation on the set \[ \{1,2,3\}. \]

Hint:

To build an equivalence relation from a minimal set, sequentially satisfy reflexivity first, then symmetry, and lastly check transitivity to find the minimal required elements.

Answer 1

Concept:
• A relation is symmetric if \( (a,b) \in R \implies (b,a) \in R \).
• An equivalence relation must be reflexive, symmetric, and transitive simultaneously.

Step 1: Check for symmetry in R.
We are given \( R = \{(1, 1), (3, 3), (1, 2)\} \) on set \( A = \{1, 2, 3\} \). Notice that the ordered pair \( (1, 2) \in R \), but its reverse pair \( (2, 1) \notin R \). Therefore, \( R \) is not symmetric.

Step 2: Identify the requirements to make \( R \cup R_1 \) an equivalence relation.
Let the equivalence relation be \( E = R \cup R_1 \).
• Reflexivity Requirement: For \( E \) to be reflexive, it must contain \( (a,a) \) for every element \( a \in A \). Thus, we must have \( (1,1), (2,2), (3,3) \in E \). Since \( (1,1) \) and \( (3,3) \) are already in \( R \), we must add \( (2,2) \) via \( R_1 \).
• Symmetry Requirement: Since \( (1,2) \in R \), its reverse pair \( (2,1) \) must be included in \( E \), meaning \( (2,1) \) must be added via \( R_1 \).
• Transitivity Requirement: Let's check pairs with the new additions: \( \{(1,1), (2,2), (3,3), (1,2), (2,1)\} \). Here \( (1,2) \) and \( (2,1) \implies (1,1) \), and \( (2,1) \) and \( (1,2) \implies (2,2) \). The set is already transitive.

Step 3: Construct the smallest relation set \( R_1 \).
The missing essential pairs that must be provided by \( R_1 \) are: \[ R_1 = \{(2, 2), (2, 1)\} \]