A problem related to Relation notation.

Question

Consider number $2$ $and$ $3$. There is a relation between these numbers that later one (3) is just 1 greater then first one (2). I can represent this sentence like this: $3$ R $2$, Where R mean "greater then" . Now there is another notation like (3,2) ∈ R. [Here in ordered pair first element is greater then second element]. Here I having problem. In first notation R simply mean "greater then" but in second notation we are considering R to be a set. How could "greater then" be a set? Pardon me if this seems stupid. But I would be grateful if you consider explaining where I am doing mistake. Thanks

Answer 1

Subsets of a set correspond with properties.

For instance if we look at $\mathbb Z$ then the fact that e.g. $5$ is a positive integer can be expressed by: $5\in\{n\in\mathbb Z\mid n>0\}$ showing that the property of "being positive" corresponds with the subset of positive elements.

"How could "greater then" be a set?"

If it comes to relations then we are looking at properties of ordered pairs.

The statement that $3$ is greater than $2$ can be looked at as a property of ordered pair $(3,2)$.

Denoting this property by $P$ we then have the set $\{(n,m)\in\mathbb Z\times\mathbb Z\mid P(n,m)\}$ and denoting this set with $R$ for $(n,m)\in\mathbb Z\times\mathbb Z$ we have:$$P(n,m)\iff(n,m)\in R$$

So in this context $(3,2)\in R$ means exactly that $3$ is greater than $2$.

Further $3R2$ must be seen as nothing more or less than an abbreviation of $(3,2)\in R$.

Answer 2

If your textbook (or whatever reference you are using) talks about "relations" it should have a definition of the word "relation". What is that definition. A common one in math texts is "a relation on set X is a subset of XxX (the set of ordered pairs of elements of X)". That is, the standard mathematical definition of "relation" is as a set! a has relation "R" to b if and only if (a, b) is in that set.