Relations §

Relations are a fundamental concept in mathematics, particularly in set theory, algebra, and discrete mathematics. They describe how elements from one set are connected to elements of another set (or the same set). Understanding relations is crucial for analyzing mathematical structures, functions, graphs, and various properties of algebraic systems.

Definition §

A relation $R$ from a set $A$ to a set $B$ is a subset of the Cartesian product $A\times B$. That is, it consists of ordered pairs $(a,b)$ where $a\in A$ and $b\in B$. If $A=B$, we say that $R$ is a relation on $A$.

Example §

If $A=\{1,2,3\}$ and $B=\{x,y\}$, an example of a relation from $A$ to $B$ could be $R=\{(1,x),(2,y),(3,x)\}$.

Representing Relations §

Relations can be represented in several ways:

• Graphically, as a set of points in a Cartesian plane or through a directed graph (digraph), where nodes represent elements of the sets and edges represent the relations.
• As a Matrix, where the matrix’s entries indicate whether the row element is related to the column element.
• Through a Table, listing the related pairs.

Characteristics of Relations §

Relations can have various properties, including:

Reflexivity §

A relation $R$ on a set $A$ is reflexive if every element is related to itself, i.e., $\forall a\in A,(a,a)\in R$.

Symmetry §

A relation $R$ on a set $A$ is symmetric if whenever an element $a$ is related to an element $b$, then $b$ is also related to $a$, i.e., $\forall a,b\in A,(a,b)\in R⟹(b,a)\in R$.

Anti-Symmetry §

A relation $R$ on a set $A$ is anti-symmetric if whenever $a$ and $b$ are distinct and $a$ is related to $b$, then $b$ is not related to $a$, i.e., $\forall a,b\in A,(a,b)\in R\wedge (b,a)\in R⟹a=b$.

Transitivity §

A relation $R$ on a set $A$ is transitive if whenever an element $a$ is related to an element $b$, and $b$ is related to an element $c$, then $a$ is also related to $c$, i.e., $\forall a,b,c\in A,(a,b)\in R\wedge (b,c)\in R⟹(a,c)\in R$.

Diagrams §

Visualizing relations using diagrams helps in understanding the connection between elements. For instance, in a directed graph representing a relation, nodes correspond to elements of the set, and directed edges show how elements are related according to the relation.

Test Questions §

1. [Basic] What is a relation in mathematics? Back: A relation is a subset of the Cartesian product of two sets, consisting of ordered pairs that describe how elements of one set are connected to elements of another.

2. [Basic] Explain the difference between symmetric and anti-symmetric relations. Back: A relation is symmetric if for all pairs $(a,b)$ in the relation, $(b,a)$ is also in the relation. It is anti-symmetric if, whenever both $(a,b)$ and $(b,a)$ are in the relation, then $a$ must be equal to $b$.

3. [Basic] Describe the conditions under which a relation is considered transitive. Back: A relation is transitive if, whenever an element $a$ is related to $b$, and $b$ is related to $c$, then $a$ is also related to $c$.

Understanding relations and their properties is vital for deeper insights into mathematical concepts and their applications in various fields.