Understanding Relation Properties on a Set

To determine which relation among the options is symmetric but neither reflexive nor transitive for the given set \(A = \{a, b, c\}\), we need to analyze the properties of relations: reflexivity, symmetry, and transitivity.

Reflexive Relation Definition

A relation \(R\) on a set \(A\) is considered reflexive if every element in the set is related to itself. Mathematically, for every element \(x \in A\), the ordered pair \((x, x)\) must be present in the relation \(R\). For the set \(A = \{a, b, c\}\), a reflexive relation would necessarily include \((a, a)\), \((b, b)\), and \((c, c)\).

Symmetric Relation Definition

A relation \(R\) on a set \(A\) is considered symmetric if whenever an element \(x\) is related to an element \(y\), then \(y\) must also be related to \(x\). Mathematically, if \((x, y) \in R\), then it must imply that \((y, x) \in R\) for all \(x, y \in A\).

Transitive Relation Definition

A relation \(R\) on a set \(A\) is considered transitive if whenever an element \(x\) is related to an element \(y\), and \(y\) is related to an element \(z\), then \(x\) must also be related to \(z\). Mathematically, if \((x, y) \in R\) and \((y, z) \in R\), then it must imply that \((x, z) \in R\) for all \(x, y, z \in A\).

Analyzing the Given Relation

Let's consider the relation \(R = \{(a, b), (b, a), (b, c), (c, b)\}\) from the options and check its properties against the definitions for the set \(A = \{a, b, c\}\).

1. Checking for Reflexivity

• For \(R\) to be reflexive on set \(A = \{a, b, c\}\), it must contain the pairs \((a, a)\), \((b, b)\), and \((c, c)\).
• Upon inspecting \(R\), we observe that \((a, a) \notin R\), \((b, b) \notin R\), and \((c, c) \notin R\).
• Therefore, the relation \(R\) is not reflexive.

2. Checking for Symmetry

• We need to check if for every pair \((x, y)\) in \(R\), its inverse pair \((y, x)\) is also in \(R\).
• For \((a, b) \in R\), we check if \((b, a) \in R\). Yes, \((b, a)\) is present in \(R\).
• For \((b, a) \in R\), we check if \((a, b) \in R\). Yes, \((a, b)\) is present in \(R\).
• For \((b, c) \in R\), we check if \((c, b) \in R\). Yes, \((c, b)\) is present in \(R\).
• For \((c, b) \in R\), we check if \((b, c) \in R\). Yes, \((b, c)\) is present in \(R\).
• Since all conditions for symmetry are met, the relation \(R\) is symmetric.

3. Checking for Transitivity

• We need to check if for every \((x, y) \in R\) and \((y, z) \in R\), it implies that \((x, z) \in R\).
• Consider the pairs \((a, b) \in R\) and \((b, c) \in R\). For \(R\) to be transitive, \((a, c)\) must be in \(R\).
• However, \((a, c)\) is not present in \(R\).
• This single instance is enough to conclude that the relation is not transitive. Let's also check another path:
• Consider the pairs \((c, b) \in R\) and \((b, a) \in R\). For \(R\) to be transitive, \((c, a)\) must be in \(R\).
• However, \((c, a)\) is not present in \(R\).
• Therefore, the relation \(R\) is not transitive.

Conclusion for the Relation

Based on our analysis, the relation \(R = \{(a, b), (b, a), (b, c), (c, b)\}\) satisfies the following conditions:

• It is symmetric.
• It is not reflexive.
• It is not transitive.

This aligns perfectly with the requirements stated in the question: "symmetric but neither reflexive nor transitive".