6 - Binary Relations

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.