Question

Give an example of a binary relation which is a) Reflexive and symmetric, but not transitive; b) Reflexive, but neither symmetric nor transitive; c) Symmetric, but neither reflexive nor transitive; d) Transitive, but neither reflexive nor symmetric.

Short answer

Examples are: a) \( \{(1,1), (2,2), (3,3), (1,2), (2,1)\} \); b) \( \{(1,1), (2,2), (1,2)\} \); c) \( \{(1,2), (2,1)\} \); d) \( \{(1,2), (2,3), (1,3)\} \).

Step-by-step solution

Understanding the Definitions

Before we find examples, let's understand the definitions: Reflexive means any element is related to itself, symmetric means if an element is related to another, than the other is related to the first, and transitive means if an element relates to a second and this second relates to a third, then the first relates to the third.

Example for Reflexive and Symmetric but Not Transitive

Consider the set \( A = \{1, 2, 3\} \) with the relation \( R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\} \). This relation is reflexive because every element is related to itself (1 to 1, 2 to 2, 3 to 3) and symmetric because \( (1,2) \) implies \( (2,1) \) is also present. However, it is not transitive because \( (1,2) \) and \( (2,1) \) do not imply \( (1,1) \) and \( (2,2) \) do not imply \( (1,2) \).

Example for Reflexive but Neither Symmetric nor Transitive

Consider the set \( A = \{1, 2\} \) with the relation \( R = \{(1,1), (2,2), (1,2)\} \). This is reflexive because \( 1 \) is related to \( 1 \) and \( 2 \) is related to \( 2 \). However, it is not symmetric because \( (1,2) \) does not imply \( (2,1) \). It is not transitive because \( (1,2) \) and \( (2,2) \) do not imply \( (1,2) \) since \( (1,2) \) is already explicit and there is no \( (2,1) \).

Example for Symmetric but Neither Reflexive nor Transitive

Consider the set \( A = \{1, 2\} \) with the relation \( R = \{(1,2), (2,1)\} \). This relation is symmetric because \( (1,2) \) also includes \( (2,1) \). It is not reflexive because \( 1 \) and \( 2 \) are not related to themselves. It is not transitive because there is no triple relation like \( (1,2), (2,1), (1,1) \).

Example for Transitive but Neither Reflexive nor Symmetric

Consider the set \( A = \{1, 2, 3\} \) with the relation \( R = \{(1,2), (2,3), (1,3)\} \). This is transitive because \( (1,2) \) and \( (2,3) \) imply \( (1,3) \). However, it is not reflexive since 1, 2, and 3 are not related to themselves. It is also not symmetric since \( (1,2) \) does not imply \( (2,1) \) and similarly for the other pairs.

Key concepts

Reflexive Relation

A binary relation over a set is called **reflexive** if every element in the set is related to itself. This fundamental concept ensures that for every element \( a \) in a set \( A \), the relation \( (a,a) \) must be present. To visualize this, imagine that each element in the set has a loop connecting it to itself. Reflexivity establishes these self-connections.

**Characteristics of Reflexive Relations**

• Each element is linked to itself. For example, in a set \( \{1, 2, 3\} \), a reflexive relation would include \{(1,1), (2,2), (3,3)\}.
• Even if other elements are related, the self-connection remains a priority.

**Examples and Non-examples**
Take the relation \( R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\} \) on the set \( A = \{1, 2, 3\} \). Here, it is reflexive because each element (1, 2, and 3) includes a self-loop, establishing reflexivity without considering other pairs.

Symmetric Relation

A binary relation is termed **symmetric** if, whenever one element is related to another, the second element is also related to the first. Essentially, symmetry assures that the direction of the relationship does not matter – once a pair is established, its inverse must also be present.

**Key Traits of Symmetric Relations**

• If \( (a,b) \) is in the relation, then \( (b,a) \) must also be included.
• It does not have conditions on self-pairing unless reflexivity is also required.

**Illustrative Example**
Consider a set \( A = \{1, 2\} \) with the relation \( R = \{(1,2), (2,1)\} \). This is symmetric because the pair \( (1,2) \) implies the existence of \( (2,1) \). Note that no self-pairing like \( (1,1) \) or \( (2,2) \) is necessary for symmetry, although reflexivity would require them.

**Lack of Symmetry in a Relation**
A relation \( R = \{(1,2)\} \) alone on a set \( A = \{1, 2\} \) would not be symmetric because \( (2,1) \) is missing. Symmetry demands the inclusion of both directions.

Transitive Relation

The notion of a **transitive relation** requires that if an element relates to a second, and this second relates to a third, then the first must relate to the third. This creates a chain-like effect among related elements, ensuring cohesive pathways within the relation.

**Recognizing Transitive Relations**

• If \( (a,b) \) and \( (b,c) \) are part of the relation, then \( (a,c) \) must also be included for it to be transitive.
• Transitivity often involves examining these connections, forming chains of two or more relationships.

**Example of Transitive Relation**On the set \( A = \{1, 2, 3\} \), the relation \( R = \{(1,2), (2,3), (1,3)\} \) is transitive. It reveals chain linking: since \( (1,2) \) and \( (2,3) \) are in the relation, it naturally implies \( (1,3) \) must be there too.

**Without Transitivity**A relation like \( R = \{(1,2), (2,3)\} \) on the set \( A = \{1, 2, 3\} \) is not transitive since \( (1,3) \) is required to complete the pathway but is not included. Transitivity ensures continuity in mapping pairs.