Definition of Relation of a Set

Question

The standard definition of a relation of an arbitrary set A is a subset of the set product of A, AxA. Is it okay to define relation R to be a subset of the set product AxA such that R has at least one property P (i.e. inequality, equality, difference, etc.), where any two element of a relation obeys P, and any two element of a set A either obeys P or not? The standard definition of a relation seems to me little ambiguous.

Answer 1

Well I personally disagree with your sense of ambiguity that you feel. I think this definition perfectly encapsulates the intuitive notion of a relation. For example, for $a, b \in A$, the statement "$a$ is related to $b$" can be written in shorthand as $(a,b)$. A "relation" is then a specific way of grouping these related pairs together, so it seems intuitive to me to make $R \subseteq A \times A$ containing these $(a, b)$'s. Your addition specification of a property doesn't seem necessary to me.

Answer 2

A relation is designed to be as generic a thing as possible, hence why it's just a set of pairs of elements. It's possible to create a rule, or identify a property, that defines the relation, but that's just a shorthand so you don't always have to write out all the pairs manually (which is particularly a pain when the relation isn't finite). It's like how a set is just a collection of objects (with a few caveats), so that $\{0, 1, 2, \mbox{cat}, \mbox{dog}, \mbox{eggplant emoji}, \emptyset, \pi, \{3+2i\}, \mbox{a lingering sense of dread}\}$ is as valid a set as $\mathbb{C}$ or $\mathbb{N}$ or $\{x \in \mathbb{R} | \sin{(x)} = 0\}$.

Answer 3

$R$ is a relation over the set $A$, if and only if, $R$ is a subset of the Cartesian square of $A$.   $$R\subseteq A{\times}A$$

That is unambiguous.   All possible subsets of $A^2$ are each a relation over $A$.

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Now we can describe some relations by set constructions when given some identified predicate, $P$. $$R=\{(a,b)\in A^2: P(a,b)\}$$

But there's a bit of chicken-and-egg redundancy there, and often there's no readily identifiable predicate other than asserting that the pair is in the given subset, $R$.

Answer 4

I think what you're trying to do could be illustrated in the following example, here's a way to define a relation:

Let $ \sim $ be a relation over $\mathbb{Z}$ defined by $$ x \sim y \iff (x=y) \lor (x+y=3).$$

In this case, $\sim$ represents $R$ and, by giving the appropriate definition, you can specify which elements are in the relation. In my example $(x,y) \in R$ (or $x \sim y$) if and only if $x$ and $y$ meet the chosen definition. Of course '$\sim$' could be any symbol you choose.

Answer 5

Your intuition is correct, but it's not necessary to add another part to it since what you're suggesting is already implied by the definition of a Relation of a Set.

A relation describes what elements in a Cartesian product are related to each other. Say we have set $A$ and set $B$, then $A \times B$ gives us all the possible ordered pairs resulting from the cross product.

Within the set of ordered pairs in $A \times B$, lies a subset that consists of elements which are $\mathbf{related\ in\ some\ way\ by\ a\ certain\ condition}$ (this is our relation $R$ from $A$ to $B$).

It makes sense that $R$ is a subset of $A \times B$, because all the ordered pairs that fall within $R$ are the ordered pairs from $A \times B$ that either meet or don't meet the condition given by $R$.

I believe the definition that you are trying to add is already implied by the definition of a relation. The only way to $\mathbf{distinguish}$ whether an ordered pair is an element of $R$, is to test that pair against a condition given by $R$.

Here's a quick example:

Let, $A = \{2 , 6, 5 \}$ and $B = \{3, 7, 8 \}$ and $R$ be a relation from $A$ to $B$, and given any ordered pair $(a , b) \in A \times B$

Our condition is $( a , b ) \in R$ iff $a \lt b$

Ok, so we have a condition that an ordered pair must meet to fall into the relation: $a \lt b$ (This is what you were adding in your definition)

First, here's our $A \times B$:

$ A \times B = \{ (2,3) , (2,7) , (2,8), (6,3), (6,7), (6,8) , (5,3) , (5,7) , (5,8) \}$

Here's a few examples showing which ordered pairs meets the relation $R$ condition:

$2 \lt 3$ , so $(2 , 3) \in R$

$6 \not\lt 3$, so $( 6 , 3 ) \notin R$

... and so on giving us:

$R = \{ (2,3) , (2,7) , (2,8) , (6,7), (6,8), (5,7) , (5,8) \}$

Here's the key: all the elements that fall within our relation $R$ are $\mathbf{contained\ in}$ $A \times B$ and are the ordered pairs that $\mathbf{meet\ a\ certain\ condition}$.

Answer 6

A relation on $A$ is as such defined to be a subset of $A\times A$.

Defining a relation on $A$ is to explicitly enumerate which elements of $A\times A$ that is included in the relation or determine the relation implicitly as those elements in $A\times A$ which are related by some predicate $P(x,y)$.

I think you mix up the definitions of the term relation with the definition of a specific relation.