On wikipedia in the section 'Generalization to binary relations' there is the following:
If $R$ is an arbitrary binary relation on $X\times Y$, the set $\{ y\in Y |\: xRy, x\in X \}$ is called the image, or the range, of $R$. Dually, the set $\{x\in X |\: xRy, y\in Y \}$ is called the domain of $R$.
This set builder notation is kind of confusing. Can anyone give a concrete example of a binary relation and show what is meant by this article?
On
Say we define $R$ on $\mathbb{R}\times \mathbb{R}$ by $$(x,y)\in R\iff y^2=x$$
Then the range is $\mathbb{R}$ and domain is $\mathbb{R}^+$.
On
The notation you gave is a bit ambiguous because there are no quantifiers:
Suppose $R \subseteq X \times Y$. Instead of $(x,y) \in R$ we can also write $xRy$; those are synonyms.
Then $y \in Y$ is in the range of $R$ iff $y$ occurs as the second element in some pair: $\exists x \in X: xRy$. We could also write that the range is $\pi_Y[R]$, where $\pi_Y: X \times Y \to Y$ is the usual projection $\pi_Y(x,y)=y$.
The domain is just $\pi_X[R]$, the other projection of $R$, or all $x \in X$ such that $\exists y \in Y: xRy$. Note the reversal of roles of $x$ and $y$ compared to the range here. If we have $R \subseteq X \times Y$ we also have $R^{-1} \subseteq Y \times X$ defined by $\{(y,x) \in Y \times X: (x,y) \in R\}$, so we swap all pairs of $R$ around. Then $\text{Range}(R^{-1})=\text{Domain}(R)$ and $\text{Domain}(R^{-1})=\text{Range}(R)$.
A simple example $X=Y=\{1,2\}$ and $R=\{(1,1),(1,2),(2,2)\}$ then the range is $Y$ as $1$ occurs in $(1,1)$ as a second element and $2$ in $(1,2)$. The domain is $X$ as $1$ and $2$ occur both as first elements. The same $R$ is also a relation on $X=\{1,2,3\}, Y=\{0,1,2\}$ and the domain and range stay the same of course. So these can be proper subsets of $X$ resp. $Y$. For the original $X,Y$ $xRy$ iff $x \le y$. (orders are a typical case of relations, as are functions:)
A function $f: X \to Y$ is just a relation $R \subseteq X \times Y$ with two extra conditions:
$\text{Domain}(f)=X$ and $xRy$ and $xRy'$ implies $y=y'$ for all $x \in X,y,y'\in Y$ (the "image" of $x$ is uniquely determined). The example $R$ I gave above is not a function as both $1R1$ and $1R2$ hold.
On
If $R$ is an arbitrary binary relation on $X\times Y$, the set $\{ y\in Y |\: xRy, x\in X \}$ is called the image, or the range, of $R$. Dually, the set $\{x\in X |\: xRy, y\in Y \}$ is called the domain of $R$.
This set builder notation is kind of confusing. Can anyone give a concrete example of a binary relation and show what is meant by this article?
Yes, those builds should have existential quantifiers.
The Image of $\mathrm R$, where $\mathrm R\subseteq X{\times}Y$, is the set of elements in $Y$ (the codomain) which are mapped by some element in $X$ (the domain). It is the set of every element which exists as the second member for some pair in $\mathrm R$. It is the subset of the codomain which contains everything that is $\mathrm R$-related to by the range. $$\{y\in Y\mid \exists x\in X~.\langle x,y\rangle \in\mathrm R\}$$
The Range of $\mathrm R$, where $\mathrm R\subseteq X{\times}Y$, is the set of elements in $X$ (the domain) which map to some element in $Y$ (the codomain). It is the set of every element which exists as the first member for some pair in $\mathrm R$. It is the subset of the domain which contains everything that $\mathrm R$-relates to the image. $$\{x\in X\mid \exists y\in Y~.\langle x,y\rangle \in\mathrm R\}$$
Example
Let $X=\{0,1,2,3\}$, $Y=\{4,5,6,7\}$ and $\mathrm R=\{\langle0,4\rangle,\langle1,5\rangle,\langle1,6\rangle,\langle3,6\rangle\}$.
The Image is $\{4,5,6\}$, and the range is $\{0,1,3\}$.
A binary relation is one of the most fundamental structures that other concepts such as functions, equivalence relations can be defined with.
A binary relation denoted $R$ over two sets $A,B$ is a subset of the Cartesian product of $A$ and $B$,e.g.$$R⊆A×B$$ Where $A×B$ is defined as :$$\left\{\left(a,b\right):a∈A \;\;\text{and}\;\; b∈B\right\}$$
and $\left(a,b\right)$ is called an ordered pair , because the order of these two elements is important, in other words for $a≠b$,$$\left(a,b\right)≠\left(b,a\right)$$
Let me give you an example:
Assume the two sets $A$ and $B$ are defined as: $$A:=\left\{1,2\right\}$$ $$B:=\left\{3,4,5\right\}$$ and $R$ is a subset of $\left\{1,2\right\}×\left\{3,4,5\right\}$
And is defined as : $R=\left\{\left(1,3\right),\left(2,5\right)\right\}$
As you see the elements of $R$ are all ordered pairs , we say $1$ is related to $3$ if and only if $1R3$ and $2$ is related to $5$ if and only if $2R5$ , the image of $R$ is by definition: $$\left\{b∈B:aRb,\ a∈A\right\}$$
So what would be the image of $R$?, just use the definition and you will see the image of $R$ is $\left\{3,5\right\}$.
Again use the definition of the domain of a binary relation:
$$\left\{a\in A:aRb,\ b\in B\right\}$$ I leave it to you to determine the domain of $R$.