6 - Binary Relations

Study Notes

Relations in Set Theory

A binary relation $R$ on a set $A$ is a set of ordered pairs $(a, b)$ where $a$ and $b$ are elements of $A$.

Reflexive Relation

A binary relation $R$ on set $A$ is reflexive if for every element $a$ in $A$, the pair $(a, a)$ is in $R$.
Example: The relation $R_0 = {(1,1), (2,2), (3,3), (4,4)}$ is reflexive because every element in the set {1, 2, 3, 4} has a pair with itself in the relation.

Anti-reflexive Relation

A binary relation $R$ on set $A$ is anti-reflexive if for every element $a$ in $A$, the pair $(a, a)$ is not in $R$.
Example: The relation $R_2 = {(1,1), (3,3)}$ is anti-reflexive because it does not contain pairs where both elements are the same (e.g., (2,2), (4,4)).

Symmetric Relation

A binary relation $R$ on set $A$ is symmetric if whenever $(a, b)$ is in $R$, then $(b, a)$ is also in $R$.
Example: The relation $R_3 = {(1,1), (1,2), (2,1), (2,4), (4,2)}$ is symmetric because if there is a relationship between two elements, the relationship also exists in reverse.

Non-Symmetric Relation

A binary relation $R$ on set $A$ is not symmetric if there exists a pair $(a, b)$ in $R$, but $(b, a)$ is not in $R$.
Example: The relation $R_4 = {(1,1), (4,3)}$ is not symmetric because while the pair (4, 3) is in the relation, the pair (3, 4) is not.

Diagrammatic Representation

The document also uses diagrams with arrows to illustrate relationships.
An arrow from point $a$ to point $b$ indicates that the pair $(a, b)$ is in the relation.

Note

The examples in the document are all based on the set {1, 2, 3, 4}.
Tables and diagrams visualize the relations, making them easier to understand.
The key concepts are reflexivity, anti-reflexivity, and symmetry.

Description

Test your understanding of binary relations in set theory, focusing on reflexive, anti-reflexive, and symmetric relations. This quiz will assess your ability to identify and provide examples of each type of relation with ordered pairs. Prepare to demonstrate your knowledge of fundamental concepts in set theory!