A binary relation is a relation between two objects. If $x$ is related to $y$ under the relation $\mathbb{R}$, then we write $_x\mathit{R}_y$
Thus, $\mathbb{R}=\{(x,y): x\in \text{A, } y \in \text{B } \text{ and } _x\mathit{R}_y\}$ where $\text{A}$ and $\text{B}$ are two sets and a relation $\mathbb{R}$ from $\text{A}$ to $\text{B}$ is the subset of the cartesian product $\text{A} \times \text{B}$.
$\square $ Domain of the relation – The set of first entries of the ordered pairs in a relation.
$\square $ Range of the relation – The set of second entries of the ordered pairs in a relation.
$\bigcirc$ Question :
- Suppose we have three sets $\text{A}$, $\text{B}$ and $\text{C}$, then how do we define a relation between them and what would be the domain and range of that relation?
- What would be the domain and range of the relation, where we have the cartesian product of more than three sets?
I don't have a strong logic but I think –
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it is obvious that no matter how many sets are there, from the ordered pair of their cartesian product the set of first entries will always be the domain of a relation between those sets (if such a relation exists). But I have no ideas, what to call the second and third entries in each ordered pair.
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my first question is related to composite mapping. If we say domain and range – the relation should be always binary [such as $\text{A}\to \text{B}$, then $(\text{A}\circ \text{B})\to \text{C}$ ]
Any explanation is valuable and highly appreciated.
Best Answer
As a subset of A×B×C.
It is a collection of triplets.
Domain, codomain and range are meaningless.
Instead one uses the 1st, 2nd, and 3rd projections.
As a subset of the product of the n sets.
It is a collection of n-tuplets.
Again, domain, codomain and range are meaningless.
Instead one uses the projections, the first through the n-th.