Relations

Let there be two sets $A$ and $B$. Then, any subset of a Cartesian product $A×B$ of two sets $A$ and $B$ is called a relation.

A relation from a set $A$ to a set $B$ is denoted by $x ℜ y$, if $x∈A$ and $y∈B$, or simply by $ℜ$ if $(x, y)∈ℜ$. A relation from a set $A$ to itself is called a relation on $A$.

Let $A=\{1, 2, 3\}$ and $B=\{1, 2, 3, 4\}$, then,

$A×B=\{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2),$$(2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)\}$

Then, a relation $ℜ$ from $A$ to $B$ be described by $ℜ=\{(x, y): x=y\}$ is $ℜ=\{(1, 1), (2, 2), (3, 3)\} ⊂ A×B$.

Another example, $ℜ_1=\{(x, y):x<y\}$ is $ℜ_1=\{(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)\}$$⊂ A×B$.

In tabular form, the above relations can be represented as;

Tabular form representing relations

In graph, the above relations can be represented as;

Graph representing relations

Domain and Range

The domain of a relation $ℜ$ is the set of all first members of the pairs $(x, y)$ of $ℜ$. It is denoted by $\text{Dom}(ℜ)$.

\[\text{Dom}(ℜ)=\{x:(x, y)∈ℜ\;\text{for some}\;y∈B\}\]

The range of a relation $ℜ$ is the set of all second members of the pairs $(x, y)$ of $ℜ$. It is denoted by $\text{Ran}(ℜ)$.

\[\text{Ran}(ℜ)=\{y:(x, y)∈ℜ\;\text{for some x∈A}\}\]

Example 1. \[ℜ=\{(1, 1), (2, 2), (3, 3)\}\]

\[\text{Dom}(ℜ)=\{1, 2, 3\}\;\text{and}\;\text{Ran}(ℜ)=\{1, 2, 3\}\]

Example 2. \[ℜ_1=\{(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)\}\]

\[\text{Dom}(ℜ_1)=\{1, 2, 3\}\;\text{and}\;\text{Ran}(ℜ_1)=\{2, 3, 4\}\]

Arrow diagrams representing relations

Inverse Relation

Since a relation is a subset of a Cartesian product, a set can be formed by interchanging the first and second members of a relation. Thus, every relation from a set $A$ to a set $B$ has a relation from $B$ to $A$. Suppose a relation,

\[ℜ=\{(x, y):x∈A, y∈B\}⊂A×B\]

then, a relation of the form,

$\{(y, x):y∈B, x∈A\}⊂B×A$ is called inverse relation from $B$ to $A$ of $ℜ$. This relation is denoted by $ℜ^{-1}$.

Let $A=\{1, 2, 3\}$ and $B=\{a, b\}$ be two sets. Then,

$A×B=\{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\}$

and, $B×A=\{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$

If $ℜ=\{(1, a), (2, b)\}$, then $ℜ^{-1}=\{(a, 1), (b, 2)\}$

Also, $\text{Dom}(ℜ)=\{1, 2\}$ and $\text{Ran}(ℜ)=\{a, b\}$

and, $\text{Dom}(ℜ^{-1})=\{a, b\}$ and $\text{Ran}(ℜ^{-1})=\{1, 2\}$

Again, let’s consider relation on $A$,

$A×A=\{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3),$$(3, 1), (3, 2), (3, 3)\}$

Here, the elements $(1, 1)$, $(2, 2)$ and $(3, 3)$ are called the diagonal elements of $A×A$.

If $ℜ=\{(1, 1), (2, 2), (3, 3)\}$ then $ℜ^{-1}=\{(1, 1), (2, 2), (3, 3)\}$

$\text{Dom}(ℜ)=\{1, 2, 3\}=\text{Dom}(ℜ^{-1})$

$\text{Ran}(ℜ)=\{1, 2, 3\}=\text{Ran}(ℜ^{-1})$


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