Question

Show by examples that \(R\) may be (a) reflexive and symmetric, without being transitive; (b) reflexive and transitive without being symmetric. Does symmetry plus transitivity imply reflexivity? Give a proof or counterexample.

Short answer

(a) Example: \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\} \). (b) Example: \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\} \). Symmetry plus transitivity does not imply reflexivity, counterexample given.

Step-by-step solution

Understanding Reflexive, Symmetric, and Transitive

1. A relation \( R \) on a set is **reflexive** if every element is related to itself. For each \( x \) in set \( A \), \( (x, x) \in R \).2. **Symmetric** means if \( (x, y) \in R \), then \( (y, x) \in R \).3. **Transitive** means if \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \).

Finding Example for (a)

Consider the set \( A = \{1, 2, 3\} \) and the relation \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\} \).- Reflexive: Every element is related to itself.- Symmetric: \( (1, 2) \) implies \( (2, 1) \).- Not Transitive: \( (1, 2) \) and \( (2, 1) \), but not \( (1, 1) \).Thus, \( R \) is reflexive and symmetric, but not transitive.

Finding Example for (b)

Consider the set \( A = \{1, 2, 3\} \) and the relation \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\} \).- Reflexive: Every element is related to itself.- Transitive: \( (1, 2) \) and \( (2, 3) \) implies \( (1, 3) \) which isn't in \( R \), correction: the example should actually have \( (1, 3) \).- Not Symmetric: \( (1, 2) \) exists but \( (2, 1) \) does not.Thus, \( R \) is reflexive and transitive, but not symmetric.

Checking Symmetry and Transitivity Implies Reflexivity

Consider the set \( A = \{1, 2\} \) and the relation \( R = \{(1, 2), (2, 1)\} \).- Symmetric: \( (1, 2) \) gives \( (2, 1) \).- Transitive: \( (1, 2) \) and \( (2, 1) \) leads to \( (1, 1) \) and \( (2, 2) \).However, for this relation, reflexivity must be checked. Here, the inclusion of \( (1, 1) \) and \( (2, 2) \) is not automatic, so symmetry plus transitivity does not imply reflexivity by default unless one's base assumptions include all self-relations for closure.

Key concepts

Reflexive Relation

A reflexive relation is quite simple to understand. Imagine if each person in a family has a picture of themselves; that's reflexivity in relations. In mathematical terms, a relation \( R \) on a set \( A \) is reflexive if every element in \( A \) relates to itself. This means for every \( x \) in the set \( A \), the pair \( (x, x) \) must be in the relation \( R \). For example, if you have a set \( A = \{1, 2, 3\} \), then \( R = \{(1, 1), (2, 2), (3, 3)\} \) is reflexive because each element is paired with itself.
A reflexive relation ensures that all elements consider their own existence by having a direct link back to themselves. It is like a person thinking about themselves in a reflective moment.

Symmetric Relation

A symmetric relation is comparable to mutual friendship; if one person considers another a friend, then the feeling is reciprocated. In the context of mathematical relations, a relation \( R \) on a set is symmetric if for all pairs \( (x, y) \) in \( R \), the reverse pair \( (y, x) \) is also in \( R \). For example, using our previous set \( A = \{1, 2, 3\} \), if \((1, 2)\) belongs to \( R \), then for symmetry, \((2, 1)\) must also belong to \( R \).
Symmetry requires a kind of 'return gesture'. If you receive a kind act, you return it in kind. But note, a symmetric relation doesn't necessarily mean every element relates to every other in a loop, just that any paired elements relate in both directions.

• Symmetrical property exists in various places like in physical objects or even social concepts.
• The relation "is married to" in a set of people can be considered symmetric.

Transitive Relation

Transitive relations link together like a chain, passing one element's influence through to another. In mathematical terms, a relation \( R \) on a set \( A \) is transitive if whenever an element \( x \) relates to \( y \) and \( y \) relates to \( z \), then \( x \) must also relate to \( z \). This property reflects 'indirect connections' or chains of relations.
For instance, consider the set \( A = \{1, 2, 3\} \), if \((1, 2)\) and \((2, 3)\) are in the relation \( R \), then for transitivity, \((1, 3)\) should also be in \( R \). However, it's essential that all such cases hold true globally within the relation for it to be considered transitive.

• Transitive relation is like knowing someone through a friend, and then connectors like these are extended to reach further elements.
• "Is a descendant of" is an example of a transitive relation.

Thus, while reflexivity and symmetry have a more 'self-contained' element to them, transitivity connects the dots, making sure there is a comprehensive relational flow through indirect connections.