What is a relation in plain simple English?

Question

Can someone explain in simple English what a relation is? I found this definition: A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x x is from the first set and the object y y is from the second set, then the objects are said to be related if the ordered pair (x,y) ( x , y ) is in the relation. But I feel as though it is not intuitive

Answer 1

A relation is a way of describing when two things are related. Say that $P$ is the set of all points of a plane (such as $\Bbb R^2$, for instance) and that $C$ is the set of all circles in that plane. We can say that a point $p$ is related to a circle $c$ if $p\in c$. So, for instance, $(1,1)$ is related to the circle centered at $(0,0)$ with radius $\sqrt2$, but not to the circle centered at $(1,0)$ with radius $1$.

If we call $R$ to this relation, we can defined it formally as$$p\mathrel Rc\text{ if and only if }p\in c.$$

The formal definition if relation consists, in this case, in defining $R$ as the subset of $\Bbb R^2\times\{\text{circles in }\Bbb R^2\}$ which consists of all pars $(p,c)$ such that $p\in c$. And now $p\mathrel Rc$ is just a shortcut for $(p,c)\in R$.

Answer 2

Just to get an informal intuition, a relation between two sets is a way to extract pairs from them. For example, $\textit{being married}$ is a relation between the set of men and the set of women.

Answer 3

There are a few ways to think of what a relation is. Sometimes you can think of a multi-valued "function". Sometimes the relation is described by a rule eg the numbers $x$ and $y$ are in the relation if $x^2+y^2=r^2$. Sometimes a relation is described by a table, eg a list of names and addresses, and the name and address are in the relation if name lives at address, which is shown by the name and address being on the same row of the table.

Formally a relation is a set of ordered pairs, but it can take a bit of thought to understand why that is a good description.