Question

A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

• A. Reflexive only
• B. Reflexive and Transitive
• C. Symmetric and Transitive
• D. Symmetric only

Hint:

To test symmetry check pair reversal. If $(a,b)$ exists then $(b,a)$ must also exist.

Answer 1

The Correct Option is D

The problem requires us to determine the properties of the given relation \(R\) on set \(A = \{1, 2, 3\}\), where \(R = \{(1,2), (2,1), (2,2)\}\). Let's evaluate each property of relations: Reflexive, Symmetric, and Transitive.

1. Reflexive: A relation \(R\) on set \(A\) is reflexive if every element is related to itself, i.e., \((a,a) \in R\) for all \(a \in A\).

In set \(A\), we have elements 1, 2, and 3. Checking for reflexivity:

• \((1,1) \notin R\)
• \((2,2) \in R\)
• \((3,3) \notin R\)

1. Symmetric: A relation \(R\) is symmetric if \((a,b) \in R\) implies \((b,a) \in R\).

Checking for symmetry:

• \((1,2) \in R\) implies \((2,1) \in R\), which is true.
• \((2,1) \in R\) implies \((1,2) \in R\), which is true.
• \((2,2) \in R\) implies \((2,2) \in R\), which is trivially true.

1. Transitive: A relation \(R\) is transitive if whenever \((a,b) \in R\) and \((b,c) \in R\), then \((a,c) \in R\).

Checking for transitivity:

• \((1,2) \in R\) and \((2,1) \in R\), but \((1,1) \notin R\).
• No other relevant combinations exist as transitive checks.

Based on the evaluations above, the relation \(R\) is symmetric only, as it fails the reflexive and transitive properties.